diff --git a/stan/math/prim/fun/inv_logit.hpp b/stan/math/prim/fun/inv_logit.hpp
index efd69f53a34..08f2a940a7f 100644
--- a/stan/math/prim/fun/inv_logit.hpp
+++ b/stan/math/prim/fun/inv_logit.hpp
@@ -49,15 +49,14 @@ namespace math {
  * @return Inverse logit of argument.
  */
 inline double inv_logit(double a) {
-  using std::exp;
   if (a < 0) {
-    double exp_a = exp(a);
+    double exp_a = std::exp(a);
     if (a < LOG_EPSILON) {
       return exp_a;
     }
-    return exp_a / (1 + exp_a);
+    return exp_a / (1.0 + exp_a);
   }
-  return inv(1 + exp(-a));
+  return inv(1 + std::exp(-a));
 }
 
 /**
@@ -69,28 +68,46 @@ inline double inv_logit(double a) {
  */
 struct inv_logit_fun {
   template <typename T>
-  static inline auto fun(const T& x) {
-    return inv_logit(x);
+  static inline auto fun(T&& x) {
+    return inv_logit(std::forward<T>(x));
   }
 };
 
 /**
- * Vectorized version of inv_logit().
+ * Vectorized version of inv_logit() for containers containing ad types.
  *
- * @tparam T type of container
- * @param x container
+ * @tparam T type of std::vector
+ * @param x std::vector
  * @return Inverse logit applied to each value in x.
  */
-template <
-    typename T, require_not_var_matrix_t<T>* = nullptr,
-    require_all_not_nonscalar_prim_or_rev_kernel_expression_t<T>* = nullptr>
-inline auto inv_logit(const T& x) {
-  return apply_scalar_unary<inv_logit_fun, T>::apply(x);
+template <typename Container, require_ad_container_t<Container>* = nullptr,
+          require_all_not_nonscalar_prim_or_rev_kernel_expression_t<
+              Container>* = nullptr,
+          require_not_rev_matrix_t<Container>* = nullptr>
+inline auto inv_logit(Container&& x) {
+  return apply_scalar_unary<inv_logit_fun, Container>::apply(
+      std::forward<Container>(x));
 }
 
-// TODO(Tadej): Eigen is introducing their implementation logistic() of this
-// in 3.4. Use that once we switch to Eigen 3.4
-
+/**
+ * Vectorized version of inv_logit() for containers with arithmetic scalar
+ * types.
+ *
+ * @tparam T A type of either `std::vector` or a type that directly inherits
+ * from `Eigen::DenseBase`. The inner scalar type must not have an auto diff
+ * scalar type.
+ * @param x Eigen expression
+ * @return Inverse logit applied to each value in x.
+ */
+template <typename Container,
+          require_container_bt<std::is_arithmetic, Container>* = nullptr,
+          require_all_not_nonscalar_prim_or_rev_kernel_expression_t<
+              Container>* = nullptr>
+inline auto inv_logit(Container&& x) {
+  return apply_vector_unary<Container>::apply(
+      std::forward<Container>(x),
+      [](const auto& v) { return v.array().logistic(); });
+}
 }  // namespace math
 }  // namespace stan
 
diff --git a/stan/math/prim/functor/apply_scalar_unary.hpp b/stan/math/prim/functor/apply_scalar_unary.hpp
index e316d3033e5..2cbcae179aa 100644
--- a/stan/math/prim/functor/apply_scalar_unary.hpp
+++ b/stan/math/prim/functor/apply_scalar_unary.hpp
@@ -58,7 +58,7 @@ struct apply_scalar_unary<F, T, require_eigen_t<T>> {
    * @return Componentwise application of the function specified
    * by F to the specified matrix.
    */
-  static inline auto apply(const T& x) {
+  static inline auto apply(const std::decay_t<T>& x) {
     return x.unaryExpr([](auto&& x) {
       return apply_scalar_unary<F, std::decay_t<decltype(x)>>::apply(x);
     });
@@ -69,7 +69,7 @@ struct apply_scalar_unary<F, T, require_eigen_t<T>> {
    * expression template of type T.
    */
   using return_t = std::decay_t<decltype(
-      apply_scalar_unary<F, T>::apply(std::declval<T>()))>;
+      apply_scalar_unary<F, std::decay_t<T>>::apply(std::declval<T>()))>;
 };
 
 /**
@@ -83,7 +83,8 @@ struct apply_scalar_unary<F, T, require_floating_point_t<T>> {
   /**
    * The return type, double.
    */
-  using return_t = std::decay_t<decltype(F::fun(std::declval<T>()))>;
+  using return_t
+      = std::decay_t<decltype(F::fun(std::declval<std::decay_t<T>>()))>;
 
   /**
    * Apply the function specified by F to the specified argument.
@@ -114,11 +115,12 @@ struct apply_scalar_unary<F, T, require_complex_t<T>> {
    * @param x Argument scalar.
    * @return Result of applying F to the scalar.
    */
-  static inline auto apply(const T& x) { return F::fun(x); }
+  static inline auto apply(const std::decay_t<T>& x) { return F::fun(x); }
   /**
    * The return type
    */
-  using return_t = std::decay_t<decltype(F::fun(std::declval<T>()))>;
+  using return_t
+      = std::decay_t<decltype(F::fun(std::declval<std::decay_t<T>>()))>;
 };
 
 /**
@@ -157,13 +159,13 @@ struct apply_scalar_unary<F, T, require_integral_t<T>> {
  * @tparam T Type of element contained in standard vector.
  */
 template <typename F, typename T>
-struct apply_scalar_unary<F, std::vector<T>> {
+struct apply_scalar_unary<F, T, require_std_vector_t<T>> {
   /**
    * Return type, which is calculated recursively as a standard
    * vector of the return type of the contained type T.
    */
-  using return_t = typename std::vector<
-      plain_type_t<typename apply_scalar_unary<F, T>::return_t>>;
+  using return_t = typename std::vector<plain_type_t<
+      typename apply_scalar_unary<F, value_type_t<std::decay_t<T>>>::return_t>>;
 
   /**
    * Apply the function specified by F elementwise to the
@@ -174,10 +176,10 @@ struct apply_scalar_unary<F, std::vector<T>> {
    * @return Elementwise application of F to the elements of the
    * container.
    */
-  static inline auto apply(const std::vector<T>& x) {
+  static inline auto apply(const std::decay_t<T>& x) {
     return_t fx(x.size());
     for (size_t i = 0; i < x.size(); ++i) {
-      fx[i] = apply_scalar_unary<F, T>::apply(x[i]);
+      fx[i] = apply_scalar_unary<F, value_type_t<T>>::apply(x[i]);
     }
     return fx;
   }
diff --git a/stan/math/rev/fun/inv_logit.hpp b/stan/math/rev/fun/inv_logit.hpp
index 086b5f45bc9..ffef93a0256 100644
--- a/stan/math/rev/fun/inv_logit.hpp
+++ b/stan/math/rev/fun/inv_logit.hpp
@@ -31,6 +31,31 @@ inline auto inv_logit(const var_value<T>& a) {
   });
 }
 
+/**
+ * The inverse logit function for Eigen expressions with var value type.
+ *
+ * See inv_logit() for the double-based version.
+ *
+ * The derivative of inverse logit is
+ *
+ * \f$\frac{d}{dx} \mbox{logit}^{-1}(x) = \mbox{logit}^{-1}(x) (1 -
+ * \mbox{logit}^{-1}(x))\f$.
+ *
+ * @tparam T type of Eigen expression
+ * @param x Eigen expression
+ * @return Inverse logit of argument.
+ */
+template <typename T, require_eigen_vt<is_var, T>* = nullptr>
+inline auto inv_logit(T&& x) {
+  auto x_arena = to_arena(std::forward<T>(x));
+  arena_t<T> ret = inv_logit(x_arena.val());
+  reverse_pass_callback([x_arena, ret]() mutable {
+    x_arena.adj().array()
+        += ret.adj().array() * ret.val().array() * (1.0 - ret.val().array());
+  });
+  return ret;
+}
+
 }  // namespace math
 }  // namespace stan
 #endif