I will review the (conjectured but well evidenced) connection between families of L-functions and characteristic polynomials of random matrices. The canonical example connects the Riemann zeta function with unitary matrices. I will then explain some recent results pertaining to various moments of interest (both of characteristic polynomials and of L-functions). Our work has further connections to log-correlated fields and combinatorics. This is joint work with Jon Keating and Theo Assiotis.