Title: Elements of order p in Tate-Shafarevich groups Abstract: I'll discuss a result with Ariel Weiss, where we show that for every prime p there exist (absolutely simple) abelian varieties over Q with an element of order p in their Tate-Shafarevich groups. The abelian varieties are related to the Jacobian of the modular curve X_0(q), where q is a prime which is 1 mod p. These results are inexplicit, in the sense that we don't write the examples down, we only prove that they exist. If there is time, I'll also say a bit about work in progress on finding explicit examples (again for every p) among Jacobians of superelliptic curves.