All classes officially published are allowed. Some care should be taken for the description of sources of magic and the like as those are tied to the way the world works even if it’s just fluff. There are some “fixes” that can be operated on spells and abilities in order to make some subclasses more attractive if you’re willing to look into it. I’m open to discuss those changes that are needed in order for your character to be as cool as you want them to be.
Examples of changes to classes or spells are things like: arcane shots for fighters, witch bolt or crown of madness action economy issues, bumping up some spell damage but reducing it in other aspects, sun soul monk undertuning issues and so on.
Unearthed Arcana or other well known Homebrews are subject to a short vetting process. (No psionics though).
Other Homebrews are welcome, but will most probably involve a longer discussion about how it should be handled. I will usually prefer to undertune and then tune up rather than the other way around.