We consider variational problems in the setting of multiple-valued functions (with a fixed number of values) and multiple-valued maps into manifolds. In particular, for an energy minimizing map into a sphere, we prove that the interior singular set is at least of codimension three. We also construct an energy reducing flow for multiple-valued functions, which is H older continuous with respect to its L 2 norms. Some questions concerning regularity and vanishing of branch points are also addressed.