diff --git a/core/src/main/scala/cats/Functor.scala b/core/src/main/scala/cats/Functor.scala
index 5109a21b53c..852abb982d0 100644
--- a/core/src/main/scala/cats/Functor.scala
+++ b/core/src/main/scala/cats/Functor.scala
@@ -177,6 +177,19 @@ trait Functor[F[_]] extends Invariant[F] { self =>
    */
   def tupleRight[A, B](fa: F[A], b: B): F[(A, B)] = map(fa)(a => (a, b))
 
+  /**
+   * Modifies the `A` value in `F[A]` with the supplied function, if the function is defined for the value.
+   * Example:
+   * {{{
+   * scala> import cats.Functor
+   * scala> import cats.implicits.catsStdInstancesForList
+   *
+   * scala> Functor[List].mapOrKeep(List(1, 2, 3)) { case 2 => 42 }
+   * res0: List[Int] = List(1, 42, 3)
+   * }}}
+   */
+  def mapOrKeep[A, A1 >: A](fa: F[A])(pf: PartialFunction[A, A1]): F[A1] = map(fa)(a => pf.applyOrElse(a, _ => a))
+
   /**
    * Un-zips an `F[(A, B)]` consisting of element pairs or Tuple2 into two separate F's tupled.
    *
@@ -258,6 +271,7 @@ object Functor {
     def as[B](b: B): F[B] = typeClassInstance.as[A, B](self, b)
     def tupleLeft[B](b: B): F[(B, A)] = typeClassInstance.tupleLeft[A, B](self, b)
     def tupleRight[B](b: B): F[(A, B)] = typeClassInstance.tupleRight[A, B](self, b)
+    def mapOrKeep[A1 >: A](pf: PartialFunction[A, A1]): F[A1] = typeClassInstance.mapOrKeep[A, A1](self)(pf)
   }
   trait AllOps[F[_], A] extends Ops[F, A] with Invariant.AllOps[F, A] {
     type TypeClassType <: Functor[F]
diff --git a/docs/nomenclature.md b/docs/nomenclature.md
index 5b8c37dcfdc..eec6e1c1145 100644
--- a/docs/nomenclature.md
+++ b/docs/nomenclature.md
@@ -14,16 +14,17 @@ _WARNING_: this page is written manually, and not automatically generated, so ma
 
 ### Functor
 
-| Type          | Method Name  |
-| ------------- |--------------|
-| `F[A] => F[Unit]`  | `void`   |
-| `F[A] => B => F[B]`  | `as`   |
-| `F[A] => (A => B) => F[B]` | `map`   |
-| `F[A] => (A => B) => F[(A,B)]` | `fproduct`   |
-| `F[A] => (A => B) => F[(B,A)]` | `fproductLeft`   |
-| `F[A] => B => F[(B, A)]`  | `tupleLeft`  |
-| `F[A] => B => F[(A, B)]`  | `tupleRight` |
-| `(A => B) => (F[A] => F[B])` | `lift`   |
+| Type                           | Method Name    | Notes |
+|--------------------------------|----------------|-------|
+| `F[A] => F[Unit]`              | `void`         |
+| `F[A] => B => F[B]`            | `as`           |
+| `F[A] => (A => B) => F[B]`     | `map`          |
+| `F[A] => (A => A1) => F[A1])`  | `mapOrKeep`    | A1 >: A, the (A => A1) is a PartialFunction
+| `F[A] => (A => B) => F[(A,B)]` | `fproduct`     |
+| `F[A] => (A => B) => F[(B,A)]` | `fproductLeft` |
+| `F[A] => B => F[(B, A)]`       | `tupleLeft`    |
+| `F[A] => B => F[(A, B)]`       | `tupleRight`   |
+| `(A => B) => (F[A] => F[B])`   | `lift`         |
 
 ### Apply
 
diff --git a/tests/shared/src/test/scala/cats/tests/FunctorSuite.scala b/tests/shared/src/test/scala/cats/tests/FunctorSuite.scala
index 1c22cc08701..a1850cfae3b 100644
--- a/tests/shared/src/test/scala/cats/tests/FunctorSuite.scala
+++ b/tests/shared/src/test/scala/cats/tests/FunctorSuite.scala
@@ -45,6 +45,18 @@ class FunctorSuite extends CatsSuite {
     }
   }
 
+  test("mapOrKeep maps matching elements preserving structure") {
+    forAll { (l: List[Int], o: Option[Int], m: Map[String, Int]) =>
+      assert(l.mapOrKeep { case i if i % 2 == 0 => i + 1 } === l.map(i => if (i % 2 == 0) i + 1 else i))
+      assert(
+        o.mapOrKeep { case i if i > 0 => i + 1 } === (if (o.nonEmpty) if (o.get > 0) Some(o.get + 1) else o else None)
+      )
+      assert(
+        m.mapOrKeep { case v if v % 2 == 0 => v + 1 } === m.map { case (k, v) => k -> (if (v % 2 == 0) v + 1 else v) }
+      )
+    }
+  }
+
   test("tupleLeft and tupleRight tuple values with a constant value preserving structure") {
     forAll { (l: List[Int], o: Option[Int], m: Map[String, Int], i: Int) =>
       assert(l.tupleLeft(i) === (List.tabulate(l.length)(in => (i, l(in)))))