The free analogues of $U(n)$ in Woronowicz' theory \cite{w2} are the compact matrix quantum groups $\{ A_u(F)|F\in GL(n,\mathbb C)\}$ introduced by Wang and Van Daele. We classify here their irreducible representations. Their fusion rules turn to be related to the combinatorics of Voiculescu's circular variable. If $F\bar{F}\in\mathbb R I_n$ we find an embedding $A_u(F)_{red}\hookrightarrow C(\mathbb T)*_{red}A_o(F)$, where $A_o(F)$ is the deformation of $SU(2)$ studied in \cite{ban}. We use the representation theory and Powers' method for showing that the reduced algebras $A_u(F)_{red}$ are simple, with at most one trace.