In general, it’s only Int (== Int64 on a 64-bit machine), though a few functions support more general Integer arguments and it’s been slowly expanded over time … in particular, tuples of dimensions still have to be Int. This was discussed way back in 2014: accept more Integer types for dims::Int... arguments by stevengj · Pull Request #7956 · JuliaLang/julia · GitHub (see also issues that link this).
See this comment from 2014 by @StefanKarpinski and this discourse discussion: Why `Int` instead of `UInt` for indexing?