These are just my personal all-time favourite anime which I recommend you watch.
Some of them you might like some maybe not. A lot of them are mainstream and you most likely already know about them.
Action:
-BleachGreat action-packed anime with a decent storyline. Personally I found it boring at around ep70 or 80. Usually top 3 on most anime websites.
-Highschool of The DeadExcellent anime if you're into that blood and gore shit. In a nutshell: Zombies and badass highschool students.
A lot of fan service so beware of the ecchi. Loads of guns, Katana's and big breasts.
-Kill la KillNice storyline which features the development of the main character. Badass fighting scenes, a bit of drama and a lot of ecchi.
A lot-Angel BeatsI found Angel Beats to be quite an emotional anime, but at the same time action packed and humorous with equal strength. It's really sad though.
Sports-Kuroko no BasukeKnB easily secures a position in my top 10 list of favourite anime, and if you're into sports, it would do the same to you.
Outstanding story that features the development of Kagami Taiga - A highschool basketballer with big dreams.
Definitely worth watching.
-Haikyuu!!Another amazing sports anime similar to KnB. This time the anime follows a storyline about a small-in-height student with amazing physical abilities who joins a highschool with a poorly structured volleyball team and helps them climb the ranks in the tournament.
-Galactik FootballUnsure if you can actually call this an anime since it wasn't made by Japanese people but Galactik Football was definitely something a lot of people (including me)have watched and adored in the past (it last aired in 2006). If you're into Football this is definitely something for you.
Strategic combatI don't really know what to call this since it involves brains and combat.
Code GeassOne of my top 10 anime. Code Geass is a great anime that involves mecha-battles and strategy. The protagonist - Lelouch uses chess to control the battlefield and ultimately acts as the ears and eyes of his allies.
Death NoteI don't know if you can really call this strategic, but Death Note revolves around two extremely intelligent detective-like men who hide behind aliases and try to figure out who they are from scratch.
Sorry for the poor explanation, there's a lot more to this anime than what I just explained but that's just the outline.
There are many, many more anime that I can recommend but I don't want to make a massive post.
Honorable mentions:
-Mirai Nikki: Drama/Action.
-Elfen Lied: Action, Gore.
-Akame Ga Kill: Action
-Accell World: Simulator
-Log Horizon: Simulator
-Sword Art Online: Simulator
-Attack on Titan: Action/Drama.
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