diff --git a/src/solvers/smt2_incremental/construct_value_expr_from_smt.cpp b/src/solvers/smt2_incremental/construct_value_expr_from_smt.cpp
index 560df31c7c2..409f1e9997c 100644
--- a/src/solvers/smt2_incremental/construct_value_expr_from_smt.cpp
+++ b/src/solvers/smt2_incremental/construct_value_expr_from_smt.cpp
@@ -87,6 +87,18 @@ class value_expr_from_smt_factoryt : public smt_term_const_downcast_visitort
       "Unexpected conversion of function application to value expression.");
   }
 
+  void visit(const smt_forall_termt &forall) override
+  {
+    INVARIANT(
+      false, "Unexpected conversion of forall quantifier to value expression.");
+  }
+
+  void visit(const smt_exists_termt &exists) override
+  {
+    INVARIANT(
+      false, "Unexpected conversion of exists quantifier to value expression.");
+  }
+
 public:
   /// \brief This function is complete the external interface to this class. All
   ///   construction of this class and construction of expressions should be
diff --git a/src/solvers/smt2_incremental/smt_terms.cpp b/src/solvers/smt2_incremental/smt_terms.cpp
index e3022cb3160..1c81c0f9f4d 100644
--- a/src/solvers/smt2_incremental/smt_terms.cpp
+++ b/src/solvers/smt2_incremental/smt_terms.cpp
@@ -151,6 +151,74 @@ smt_function_application_termt::arguments() const
   });
 }
 
+smt_forall_termt::smt_forall_termt(
+  std::vector<smt_identifier_termt> bound_variables,
+  smt_termt predicate)
+  : smt_termt{ID_smt_forall_term, smt_bool_sortt{}}
+{
+  INVARIANT(
+    !bound_variables.empty(),
+    "A forall term should bind at least one variable.");
+  std::transform(
+    std::make_move_iterator(bound_variables.begin()),
+    std::make_move_iterator(bound_variables.end()),
+    std::back_inserter(get_sub()),
+    [](smt_identifier_termt &&bound_variable) {
+      return irept{std::move(bound_variable)};
+    });
+  INVARIANT(
+    predicate.get_sort().cast<smt_bool_sortt>(),
+    "Predicate of forall quantifier is expected to have bool sort.");
+  set(ID_body, std::move(predicate));
+}
+
+const smt_termt &smt_forall_termt::predicate() const
+{
+  return static_cast<const smt_termt &>(find(ID_body));
+}
+
+std::vector<std::reference_wrapper<const smt_identifier_termt>>
+smt_forall_termt::bound_variables() const
+{
+  return make_range(get_sub()).map([](const irept &variable) {
+    return std::cref(static_cast<const smt_identifier_termt &>(variable));
+  });
+}
+
+smt_exists_termt::smt_exists_termt(
+  std::vector<smt_identifier_termt> bound_variables,
+  smt_termt predicate)
+  : smt_termt{ID_smt_exists_term, smt_bool_sortt{}}
+{
+  INVARIANT(
+    !bound_variables.empty(),
+    "A exists term should bind at least one variable.");
+  std::transform(
+    std::make_move_iterator(bound_variables.begin()),
+    std::make_move_iterator(bound_variables.end()),
+    std::back_inserter(get_sub()),
+    [](smt_identifier_termt &&bound_variable) {
+      return irept{std::move(bound_variable)};
+    });
+  INVARIANT(
+    predicate.get_sort().cast<smt_bool_sortt>(),
+    "Predicate of exists quantifier is expected to have bool sort.");
+  set(ID_body, std::move(predicate));
+}
+
+const smt_termt &smt_exists_termt::predicate() const
+{
+  return static_cast<const smt_termt &>(find(ID_body));
+}
+
+std::vector<std::reference_wrapper<const smt_identifier_termt>>
+smt_exists_termt::bound_variables() const
+{
+  return make_range(get_sub()).map([](const irept &variable) {
+    return std::cref(static_cast<const smt_identifier_termt &>(variable));
+  });
+}
+
 template <typename visitort>
 void accept(const smt_termt &term, const irep_idt &id, visitort &&visitor)
 {
diff --git a/src/solvers/smt2_incremental/smt_terms.def b/src/solvers/smt2_incremental/smt_terms.def
index fbd46311e03..33fc99248e9 100644
--- a/src/solvers/smt2_incremental/smt_terms.def
+++ b/src/solvers/smt2_incremental/smt_terms.def
@@ -10,3 +10,5 @@ TERM_ID(bool_literal)
 TERM_ID(identifier)
 TERM_ID(bit_vector_constant)
 TERM_ID(function_application)
+TERM_ID(forall)
+TERM_ID(exists)
diff --git a/src/solvers/smt2_incremental/smt_terms.h b/src/solvers/smt2_incremental/smt_terms.h
index b8cfc4b4fa8..53faa567c5d 100644
--- a/src/solvers/smt2_incremental/smt_terms.h
+++ b/src/solvers/smt2_incremental/smt_terms.h
@@ -207,6 +207,28 @@ class smt_function_application_termt : public smt_termt
   };
 };
 
+class smt_forall_termt : public smt_termt
+{
+public:
+  smt_forall_termt(
+    std::vector<smt_identifier_termt> bound_variables,
+    smt_termt predicate);
+  std::vector<std::reference_wrapper<const smt_identifier_termt>>
+  bound_variables() const;
+  const smt_termt &predicate() const;
+};
+
+class smt_exists_termt : public smt_termt
+{
+public:
+  smt_exists_termt(
+    std::vector<smt_identifier_termt> bound_variables,
+    smt_termt predicate);
+  std::vector<std::reference_wrapper<const smt_identifier_termt>>
+  bound_variables() const;
+  const smt_termt &predicate() const;
+};
+
 class smt_term_const_downcast_visitort
 {
 public:
diff --git a/src/solvers/smt2_incremental/smt_to_smt2_string.cpp b/src/solvers/smt2_incremental/smt_to_smt2_string.cpp
index 8cfb44c31f3..246baff557d 100644
--- a/src/solvers/smt2_incremental/smt_to_smt2_string.cpp
+++ b/src/solvers/smt2_incremental/smt_to_smt2_string.cpp
@@ -96,6 +96,11 @@ std::string smt_to_smt2_string(const smt_sortt &sort)
   return ss.str();
 }
 
+struct sorted_variablest final
+{
+  std::vector<std::reference_wrapper<const smt_identifier_termt>> identifiers;
+};
+
 /// \note The printing algorithm in the `smt_term_to_string_convertert` class is
 ///   implemented using an explicit `std::stack` rather than using recursion
 ///   and the call stack. This is done in order to ensure we can print smt terms
@@ -124,6 +129,7 @@ class smt_term_to_string_convertert : private smt_term_const_downcast_visitort
   template <typename elementt>
   output_functiont make_output_function(
     const std::vector<std::reference_wrapper<const elementt>> &output);
+  output_functiont make_output_function(const sorted_variablest &output);
 
   /// \brief Single argument version of `push_outputs`.
   template <typename outputt>
@@ -153,6 +159,8 @@ class smt_term_to_string_convertert : private smt_term_const_downcast_visitort
   void visit(const smt_bit_vector_constant_termt &bit_vector_constant) override;
   void
   visit(const smt_function_application_termt &function_application) override;
+  void visit(const smt_forall_termt &forall) override;
+  void visit(const smt_exists_termt &exists) override;
 
 public:
   /// \brief This function is complete the external interface to this class. All
@@ -199,6 +207,25 @@ smt_term_to_string_convertert::make_output_function(
   };
 }
 
+smt_term_to_string_convertert::output_functiont
+smt_term_to_string_convertert::make_output_function(
+  const sorted_variablest &output)
+{
+  return [=](std::ostream &os) {
+    const auto push_sorted_variable =
+      [&](const smt_identifier_termt &identifier) {
+        push_outputs("(", identifier, " ", identifier.get_sort(), ")");
+      };
+    for(const auto &bound_variable :
+        make_range(output.identifiers.rbegin(), --output.identifiers.rend()))
+    {
+      push_sorted_variable(bound_variable);
+      push_output(" ");
+    }
+    push_sorted_variable(output.identifiers.front());
+  };
+}
+
 template <typename outputt>
 void smt_term_to_string_convertert::push_output(outputt &&output)
 {
@@ -255,6 +282,20 @@ void smt_term_to_string_convertert::visit(
   push_outputs("(", id, std::move(arguments), ")");
 }
 
+void smt_term_to_string_convertert::visit(const smt_forall_termt &forall)
+{
+  sorted_variablest bound_variables{forall.bound_variables()};
+  auto predicate = forall.predicate();
+  push_outputs("(forall (", bound_variables, ") ", std::move(predicate), ")");
+}
+
+void smt_term_to_string_convertert::visit(const smt_exists_termt &exists)
+{
+  sorted_variablest bound_variables{exists.bound_variables()};
+  auto predicate = exists.predicate();
+  push_outputs("(exists (", bound_variables, ") ", std::move(predicate), ")");
+}
+
 std::ostream &
 smt_term_to_string_convertert::convert(std::ostream &os, const smt_termt &term)
 {
diff --git a/unit/solvers/smt2_incremental/construct_value_expr_from_smt.cpp b/unit/solvers/smt2_incremental/construct_value_expr_from_smt.cpp
index d0a570277ce..9f151531441 100644
--- a/unit/solvers/smt2_incremental/construct_value_expr_from_smt.cpp
+++ b/unit/solvers/smt2_incremental/construct_value_expr_from_smt.cpp
@@ -127,6 +127,18 @@ TEST_CASE(
       smt_core_theoryt::make_not(smt_bool_literal_termt{true}),
       unsignedbv_typet{16},
       "Unexpected conversion of function application to value expression."},
+    rowt{
+      smt_forall_termt{
+        {smt_identifier_termt{"i", smt_bool_sortt{}}},
+        smt_bool_literal_termt{true}},
+      bool_typet{},
+      "Unexpected conversion of forall quantifier to value expression."},
+    rowt{
+      smt_exists_termt{
+        {smt_identifier_termt{"j", smt_bool_sortt{}}},
+        smt_bool_literal_termt{true}},
+      bool_typet{},
+      "Unexpected conversion of exists quantifier to value expression."},
     rowt{
       smt_bit_vector_constant_termt{0, 16},
       pointer_typet{unsigned_int_type(), 0},
diff --git a/unit/solvers/smt2_incremental/smt_terms.cpp b/unit/solvers/smt2_incremental/smt_terms.cpp
index 0405b5fb109..847cadd987c 100644
--- a/unit/solvers/smt2_incremental/smt_terms.cpp
+++ b/unit/solvers/smt2_incremental/smt_terms.cpp
@@ -1,11 +1,11 @@
 // Author: Diffblue Ltd.
 
-#include <testing-utils/use_catch.h>
+#include <util/mp_arith.h>
 
 #include <solvers/smt2_incremental/smt_core_theory.h>
 #include <solvers/smt2_incremental/smt_terms.h>
-
-#include <util/mp_arith.h>
+#include <testing-utils/invariant.h>
+#include <testing-utils/use_catch.h>
 
 #include <type_traits>
 
@@ -85,6 +85,82 @@ TEST_CASE("smt_termt equality.", "[core][smt2_incremental]")
     smt_bit_vector_constant_termt{12, 8});
 }
 
+template <typename term_typet>
+std::string term_description();
+
+template <>
+std::string term_description<smt_forall_termt>()
+{
+  return "forall";
+}
+
+template <>
+std::string term_description<smt_exists_termt>()
+{
+  return "exists";
+}
+
+TEMPLATE_TEST_CASE(
+  "smt quantifier terms",
+  "[core][smt2_incremental]",
+  smt_forall_termt,
+  smt_exists_termt)
+{
+  using quantifiert = TestType;
+  const smt_identifier_termt i{"i", smt_bit_vector_sortt{8}};
+  const smt_identifier_termt j{"j", smt_bit_vector_sortt{8}};
+  SECTION("Getters")
+  {
+    SECTION("One bound variable")
+    {
+      const auto predicate = smt_core_theoryt::equal(i, i);
+      const quantifiert quantifier{{i}, predicate};
+      CHECK(quantifier.get_sort() == smt_bool_sortt{});
+      const auto variables = quantifier.bound_variables();
+      CHECK(quantifier.predicate() == predicate);
+      REQUIRE(variables.size() == 1);
+      CHECK(variables[0].get() == i);
+    }
+    SECTION("Two bound variables")
+    {
+      const auto predicate = smt_core_theoryt::distinct(i, j);
+      const quantifiert quantifier{{i, j}, predicate};
+      CHECK(quantifier.get_sort() == smt_bool_sortt{});
+      const auto variables = quantifier.bound_variables();
+      CHECK(quantifier.predicate() == predicate);
+      REQUIRE(variables.size() == 2);
+      CHECK(variables[0].get() == i);
+      CHECK(variables[1].get() == j);
+    }
+  }
+  SECTION("Constructor validation")
+  {
+    cbmc_invariants_should_throwt invariants_throw;
+    SECTION("Empty variables")
+    {
+      const auto generate_error = [&]() {
+        quantifiert{{}, smt_core_theoryt::equal(i, i)};
+      };
+      REQUIRE_THROWS_MATCHES(
+        generate_error(),
+        invariant_failedt,
+        invariant_failure_containing(
+          "A " + term_description<quantifiert>() +
+          " term should bind at least one variable."));
+    }
+    SECTION("Non bool predicate")
+    {
+      const auto generate_error = [&]() { quantifiert{{i}, i}; };
+      REQUIRE_THROWS_MATCHES(
+        generate_error(),
+        invariant_failedt,
+        invariant_failure_containing(
+          "Predicate of " + term_description<quantifiert>() +
+          " quantifier is expected to have bool sort."));
+    }
+  }
+}
+
 template <typename expected_termt>
 class term_visit_type_checkert final : public smt_term_const_downcast_visitort
 {
@@ -139,6 +215,30 @@ class term_visit_type_checkert final : public smt_term_const_downcast_visitort
       unexpected_term_visited = true;
     }
   }
+
+  void visit(const smt_forall_termt &) override
+  {
+    if(std::is_same<expected_termt, smt_forall_termt>::value)
+    {
+      expected_term_visited = true;
+    }
+    else
+    {
+      unexpected_term_visited = true;
+    }
+  }
+
+  void visit(const smt_exists_termt &) override
+  {
+    if(std::is_same<expected_termt, smt_exists_termt>::value)
+    {
+      expected_term_visited = true;
+    }
+    else
+    {
+      unexpected_term_visited = true;
+    }
+  }
 };
 
 template <typename term_typet>
@@ -168,13 +268,31 @@ smt_function_application_termt make_test_term<smt_function_application_termt>()
   return smt_core_theoryt::make_not(smt_bool_literal_termt{true});
 }
 
+template <>
+smt_forall_termt make_test_term<smt_forall_termt>()
+{
+  const smt_identifier_termt identifier{"i", smt_bit_vector_sortt{8}};
+  return smt_forall_termt{
+    {identifier}, smt_core_theoryt::equal(identifier, identifier)};
+}
+
+template <>
+smt_exists_termt make_test_term<smt_exists_termt>()
+{
+  const smt_identifier_termt identifier{"i", smt_bit_vector_sortt{8}};
+  return smt_exists_termt{
+    {identifier}, smt_core_theoryt::equal(identifier, identifier)};
+}
+
 TEMPLATE_TEST_CASE(
   "smt_termt::accept(visitor)",
   "[core][smt2_incremental]",
   smt_bool_literal_termt,
   smt_identifier_termt,
   smt_bit_vector_constant_termt,
-  smt_function_application_termt)
+  smt_function_application_termt,
+  smt_forall_termt,
+  smt_exists_termt)
 {
   term_visit_type_checkert<TestType> checker;
   make_test_term<TestType>().accept(checker);
diff --git a/unit/solvers/smt2_incremental/smt_to_smt2_string.cpp b/unit/solvers/smt2_incremental/smt_to_smt2_string.cpp
index 60c74b5aad6..7f2b9575dc1 100644
--- a/unit/solvers/smt2_incremental/smt_to_smt2_string.cpp
+++ b/unit/solvers/smt2_incremental/smt_to_smt2_string.cpp
@@ -202,3 +202,45 @@ TEST_CASE(
   CHECK(
     smt_to_smt2_string(smt_set_logic_commandt{qf_bv}) == "(set-logic QF_BV)");
 }
+
+TEST_CASE("SMT forall term to string conversion", "[core][smt2_incremental]")
+{
+  const smt_identifier_termt i{"i", smt_bit_vector_sortt{8}};
+  const smt_identifier_termt j{"j", smt_bool_sortt{}};
+  SECTION("One bound variable")
+  {
+    const auto predicate = smt_core_theoryt::equal(i, i);
+    const smt_forall_termt forall{{i}, predicate};
+    CHECK(smt_to_smt2_string(forall) == "(forall ((i (_ BitVec 8))) (= i i))");
+  }
+  SECTION("Two bound variables")
+  {
+    const auto predicate =
+      smt_core_theoryt::make_or(smt_core_theoryt::equal(i, i), j);
+    const smt_forall_termt forall{{i, j}, predicate};
+    CHECK(
+      smt_to_smt2_string(forall) ==
+      "(forall ((i (_ BitVec 8)) (j Bool)) (or (= i i) j))");
+  }
+}
+
+TEST_CASE("SMT exists term to string conversion", "[core][smt2_incremental]")
+{
+  const smt_identifier_termt i{"i", smt_bit_vector_sortt{8}};
+  const smt_identifier_termt j{"j", smt_bool_sortt{}};
+  SECTION("One bound variable")
+  {
+    const auto predicate = smt_core_theoryt::equal(i, i);
+    const smt_exists_termt exists{{i}, predicate};
+    CHECK(smt_to_smt2_string(exists) == "(exists ((i (_ BitVec 8))) (= i i))");
+  }
+  SECTION("Two bound variables")
+  {
+    const auto predicate =
+      smt_core_theoryt::make_or(smt_core_theoryt::equal(i, i), j);
+    const smt_exists_termt exists{{i, j}, predicate};
+    CHECK(
+      smt_to_smt2_string(exists) ==
+      "(exists ((i (_ BitVec 8)) (j Bool)) (or (= i i) j))");
+  }
+}