Simple C*-algebras of high topological dimension were discovered by Villadsen in the mid 1990s, and have provided typically negative answers to many open problems. They have been little studied in their own right, however. In this talk I will discuss some invariants which are well-adapted to studying these algebras, and which may be thought of as recording the ratio of the algebra’s topological dimension to its ”matricial size”. As an application we obtain the analogue among separable amenable C*-algebras of McDuff’s construction of an uncountable family of mutually non-isomorphic II1 factors.