Like most french people I've grown with Anime, it's been over present on all medias since the end of the 70s.
Anime is like "Comics Culture" : super wide. You can't sum it up in a few words since there's absolutely everything in it, from awesome to awefull.
There are some specificities to it, but not that much in reality... the "japanese specific references" are superficial... in reality is very accessible.
For example, there's a ton of fantastic Sci-Fi Anime, beautiful stories, extraordinary animations... but they're not really original...
In recent years netflix has released several excellent Animes, I can make recommendations based on what's available there? If you light MMA and fighting/action series, you should love BAKI and KENGAN ASHURA
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Or you tell me what you like in general and I'll give you a list of suggestions based on that...