Abstract: I will begin by introducing the Birch and Swinnerton-Dyer conjecture in the context of abelian varieties attached to modular forms, and discuss some of the main results about it. I will then introduce Mazur's notion of visibility of Shafarevich-Tate groups and explain some of the basic facts and theorems. Cremona, Mazur, Agashe, and myself carried out large computations about visibility for modular abelian varieties of level N in J_0(N). These computations addressed the following question: If A is a modular abelian variety of level N, how much of the Shafarevich-Tate group of A is modular of level N, i.e., visible in J_0(N). The results of these computations suggest that often much of the Shafarevich-Tate group is NOT modular of level N. This suggests asking if every element is modular of level N*m, for some auxiliary integer m, and if so, what can one say about the set of such m? I will finish the talk with some new data and thoughts about this last question, which is still very much open.