How to replay a League of Legends match?

You can record games with http://www.leaguereplays.com/

The LRF file just plays with the game itself. You can browse the match, but you don't have information your character did not saw during the game.

Besides those limitations, I think it is possible to record video of this with comments.

This feature has been added as of season 7. Here is the announcement from Riot:

That brings us to where we are today. We are ready to announce that Replays are coming to the League client update this pre-season.

Here’s how it works:

• Download your replay from the End of Game Screen or Match History.
• Watch Replays from the current patch using our new Replay Mode, with the functionality you’re used to from Spectator Mode.
• Find cool moments using the new Annotated Timeline, which marks major events like kills, takedowns of towers and inhibitors, and dragon/baron kills.
• Capture video clips of your in-game moments (“Highlights”) using built-in recording functionality.
• These Highlights will be stored on your hard drive (C:\MyDocuments\LeagueofLegends\Highlights) as a .webm video file. These files can be viewed by opening them in any browser that supports HTML5, and shared to social media alongside your cat gifs.

With all of those pieces put together, you’ll be able to quickly capture, save, and share your favorite moments on video to share whenever you want.

Note that the full-game recordings (.rofl) give you all the features of spectator mode, so you can see more than your own team's perspective.

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I have the Alpha Client, so here's a screenshot from my match history:

• I have not downloaded a replay of the first match, so it shows the download button
• I have downloaded a replay of the second match, so it shows the replay button.

Replays currently expire after three days. Once a replay has expired, the button will be grayed out.

Edit: the previous version of this answer is out of date.

Revisiting this post on 2017-04-05 it seems like replay.gg no longer works. I know of people who use https://www.plays.tv/ which might be worth checking out.

Original post:

http://replay.gg/

You have to do some initial setup but after that it works automatically, sending you an email after each game. The email contains instructions on how to view the replay, which is in spectator format. You view the game using the LoL client, which means you do all the things you can when you spectate normally e.g. rewind, see the whole map, see every player's build.