Abstract:  In 2002, Dumitriu and Edelman discovered matrix models for the beta-Hermite and the beta-Laguerre ensembles for general values of $\beta.$   Prior to this development, matrix models for these ensembles were only known for $\beta =1, 2, 4.$   In this talk, I will discuss how to construct matrix-valued diffusion processes which describe the non-equilibrium situation of these static matrix models.   I will also present some results on the associated spectral measure processes and empirical eigenvalue/singular value processes.