Varieties were originally studied by comparing them with the hypersurfaces that are their generic projections--curves in the plane and surfaces in three-space, for example. In low dimensions, the fibers of these generic projections are pretty well understood, but there are serious obstructions to extending this understanding to all dimensions. I'll survey what's known, explain some examples, and present a new conjecture about these fibers. A connection with the regularity of powers of an ideal (asymptotic regularity) plays an interesting role.