Random matrices, number theory and the derivative of the characteristic polynomial

For over 50 years the connection between random matrix theory and the Riemann zeta function has been studied, allowing calculations on average values of the characteristic polynomial of random unitary matrices to inform studies of number theoretical functions.  I will give some stories and history of this connection and then look at some results on the derivative of the characteristic polynomial, in analogy with the derivative of the Riemann zeta function.