Uniformly HyperFinite (UHF) algebras are those C* algebras in which every finite subset is `near' a finite-dimensional full matrix subalgebra. This can be formalized in three different ways, all three being equivalent in the separable case. Separable UHF algebras were classified in the 1960s by Glimm and Dixmier. Dixmier asked whether three definitions are equivalent in the nonseparable case. I will give a complete answer to this question. Then I will state some even more basic questions that we could not answer. This is a joint work with Takeshi Katsura.