Recently, there has been a great deal of interest in the moments of derivatives of random matrix characteristic polynomials. One motivation for this is the study of the analogous moments of derivatives of the Riemann zeta function and other families of L-functions. In this talk, I will first go over the link between moments of families of L-functions and those of characteristic polynomials of random matrices. I will then motivate the study of moments of the derivatives of the zeta function and then give an account of the known results and conjectures in the random matrix theory case. Lastly, I will discuss recent joint work with Julio Andrade on the joint moments of derivatives of characteristic polynomials of symplectic and orthogonal matrices.