  We discuss consistency conditions for branes at orbifold singularities. The
conditions have a world-sheet interpretation in terms of tadpole cancellation
and a space-time interpretation in terms of anomalies. As examples, we consider
type II and type I branes on $C^2/Z_M$ orbifolds. We give orientifold
constructions of phases of type I or heterotic string theory, involving
branches with extra tensor multiplets, which arise when small SO(32) instantons
sit on orbifold singularities.
