Abstract: Many problems in number theory are related to the question of whether or not there are any Q-linear relations among the ordinates of the nontrivial zeros of automorphic L-functions. In general, this question is wide open. I will discuss recent progress on some special cases of this question. For instance, are the (non-real) zeros of an L-function simple? Given two L-functions, are their zeros distinct? Can an L-function simultaneously vanish at height t and height 2t?