Ulrich bundles on Del Pezzo threefolds

We prove that for any r > 1 the moduli space of stable Ulrich bundles of rank r and determinant O_X(r) on any smooth Fano threefold X of index two is smooth of dimension r^2 + 1 and that the same holds true for even r when the index is four, in which case no odd–rank Ulrich bundles exist. In particular this shows that any such threefold is Ulrich wild. As a preliminary result, we give necessary and sufficient conditions for the existence of Ulrich bundles on any smooth projective threefold in terms of the existence of a curve in the threefold enjoying special properties.