Abstract I will present some interesting examples and conjectures related to the Katz-Sarnak philosophy connecting random matrices and critical zeros in families of L-functions. A certain "symmetry flipping" phenomenon hints at a connection between ensembles of random matrices from the classical groups and zeros of automorphic L-functions constructed as functorial lifts (Rankin-Selberg, symmetric- and exterior- square, etc.) from modular forms. Also, in work in progress, I will describe a tantalizing conjectural connection between the phenomena of excess rank and repulsion of critical Hasse-Weil zeros in families of elliptic curves. Parts of earlier work are joint with S.J. Miller. The current work in progress is joint with Miller and D.K. Huynh, J.P. Keating and N.C. Snaith.