Wigner random matrices are a basic example of a Hermitian random matrix model, in which the upper-triangular entries are jointly independent.  The most famous example of a Wigner random matrix is the Gaussian Unitary Ensemble (GUE), which is particularly amenable to study due to its rich algebraic structure.  In particular, the fine-scale distribution of the eigenvalues is completely understood.  There has been much recent progress on extending these distribution laws to more general Wigner matrices, a phenomenon sometimes referred to as _universality_.  In this talk we will discuss recent work of Van Vu and myself on establishing several cases of this universality phenomenon, as well as parallel work of Erdos, Schlein, and Yau.