This review will mostly echo what others have said, and that is: play the games first, then watch the anime if you'd really like to. I had played the first two games (plus UDG) before watching this, and did so to stick to my fandom completion ratio, I suppose. I've had friends look at me funny when I say that I like Danganronpa and then start talking about the anime; when I clarify that I'm talking about the games, they usually respond by admitting that they didn't know the games existed.
Trigger Happy Havoc as a game is, well, much more well-rounded than its anime adaptation. I don't entirely blame the team behind it, as it is tremendously hard to fit as much material as said game has into thirteen episodes, but I do think they should have pushed for more. I'll go into my two main gripes with this adaptation.
1) Incredibly fast pacing. I had no issue with how most of the story was handled, but my god, the trials were breakneck. Another review has said something along the lines of 'it seems like Naegi and Kirigiri are pulling stuff out of their ass during trials', and the lack of explanation we got during investigation time just means that the trials themselves made little sense unless one already knew what was going to happen. I was watching the first trial episode beside a friend and commented that so much was left out, because, well... It was. The logic leaps needed on the viewer's part to follow what the characters had discovered, given that they hadn't been given most of the information, were ridiculous.
2) Flat characterization. Now, I understand that a lot of the bonding with characters in-game comes from School Mode- as I said, I've played the games. I'm not expecting the anime to have one-on-one scenes with Naegi and every other character to fill this void, but the fact that nothing at all replaced this (even made-up scenes with the class would have been preferable + simulated some kind of bond) was disappointing, to say the least. If you want to care about the characters as people, please do play the games. They're all pretty great.
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As I've acknowledged, a lot of the issues I had with this anime are ones that I know hinge on the limited amount of episodes allotted to fit the story into. I maintain that I can't blame the team for this, as I'm sure they did the best they could with condensing all of the material. However, I can't give the anime adaptation of THH more than a 'ehhh' and shrug of the shoulders.
(I will say, though- the voice acting in this anime is superb. Monokuma, Hifumi and Hagakure's voices drove me up the wall at first, and then I realized that they were all exactly as annoying as I would have expected. Monokuma's voice actually grew on me after a few episodes, because it fits so well.)