I've been around a lot of competitive play in the last year/year and a half in a different game which includes playing, analyzing, and coaching in the top 3-1% of players in the most popular PC game at the moment. That's not for brag, but rather to give a bit of credibility to my statement. While I LOVE Overwatch's competitive gameplay currently as it holds a lot of concepts and structure that is crucial in a game that wants an esports scene, there's 1 big problem that I have and it might not be what you think.

Replay systems. It's that simple. As it stands, teams and individuals looking for true improvement must completely rely on just grinding for aiming skill, theory crafting, or having to make in-game analysis as the game progresses. If you want a real, high-quality competitive scene, you need people who are well in tuned with the game and how it should be played.

By being able to look over replays from different perspectives and, most importantly, free camera, it gives players and teams the ability to collect information and data about the game on a much larger level. This allows:
-Players to identify their own mistakes
-Analyzing the enemy's team setup and what your potential openings/blind spots are
-Compare your team's formation to theirs, why whichever one worked, was able to work.
-Go over replays as a team to strategize for future situations that may be similar, and adjust their gameplay

I can keep going, but you get the idea. If Blizzard wants Overwatch to be a top-quality esport, replays will be essential. Otherwise pro games are just "who's got the better shot" since most players at that level understand the meta and what heroes to play and when. Is there any way we can get this implemented?

TL;DR: Give us a replay system please. :)

We are very eager to add a replay system to the game. It is something on our long-term road map and we've architected the engine in a way that will enable us to release the feature someday. To make replays a player-facing feature is a tremendous amount of work (like, a very large-scale task for us) and the same people who do that work are currently working on our high bandwith (60hz) client update rate feature. We've prioritized that work over a replay feature in the near term. That doesn't mean that we won't get to replays. It just means it's a big, hard feature that will take some time. But we're totally on board with adding it to the game.

Replays are now fully implemented into the game. Find them in your Career Profile.