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Nyaa Torrents is the largest anime tracker on the internet and is also a one-stop shopping center for Japanese media in general. It's totally open to the public but run by a staff that zealously purges inferior or rulebreaking content, so you have almost zero chance of downloading anything malicious off the site. It however mostly caters to new media, older stuff often goes unseeded unless it's very popular. For older titles you'll have to look at either BakaBT or AnimeBytes.
BakaBT is a semi-private tracker for anime/Japanese media. You can download off it all you want, but without an account you can't directly search the website for torrents (you can circumvent this with google searches, just append "BakaBT" to anything you want to find, e.g. "Cowboy Bebop BakaBT" into google). Also only staff members can upload torrents to BakaBT, so their collection is smaller and less diverse than Nyaa's, since they only upload what they consider the "best" releases for a show or manga. If you disagree with their choice you have to look elsewhere. There's also the fact that BakaBT tends to blacklist titles that are licensed in the west, to avoid getting C&D orders, so they might be lacking some mainstream titles because of that. However, BakaBT does have an extensive library of older, niche shows that are all well-seeded, which you can't find on Nyaa. They also have a very nice collection of anime OSTs and generally have decent manga collections, including official translation scans.
Lastly is AnimeBytes, a private tracker that is invite/application only. It's the best of both Nyaa and BakaBT, boasting both extreme variety in releases and plenty of seeds even for old, obscure titles. They also host virtually everything banned on BakaBT. Downside is obviously the seeding requirements cause it's a private tracker.