Blizzard have announced that Replays will soon arrive to Overwatch's Public Test Realm.
As mentioned on the Overwatch blog, the feature will be present on both PC and console versions of the game and it will allow you to save your most notable highlights and even record entire matches if you wish! All you need is a good vantage point and voila, you can rewatch them anytime you like!
Blizzard is really doing an impressive job with Overwatch, let's hope they keep it that way in the future as well.