We present some work in progress, on moduli spaces of Drinfeld shtukas. These spaces are the function field analogous to Shimura varieties. In fact they are more versatile; there are r-legged versions for any r. Tate's conjecture predicts some interesting relations between shtuka spaces and function field arithmetic. For instance, there should be a notion of modularity for the r-fold product of an elliptic curve. We verify these predictions in a few cases. This is partly joint work with Noam Elkies.