Negative moments of the Riemann zeta function

I will talk about the negative moments of the Riemann zeta function, both continuous and discrete, based on joint works with Bui, Keating and Milinovich. I will explain how to obtain progress towards a conjecture of Gonek on negative moments, when the shift of the negative zeta function is "small" (going to 0 with t). When we are integrating from T to 2T, and the shift is less than 1/(log T), we also discuss some conjectures based on random matrix theory and some heuristic number theoretic arguments which predict certain transition regimes in the asymptotic formulas.