A geometric averaging operator is any mapping which sends a function on a manifold to its integral over some family of submanifolds, the canonical examples being the X-ray and Radon transforms. For general families of submanifolds, little is known about the boundedness properties of these operators. This talk will discuss the motivations for studying such objects and recent sharp results (in terms of $L^p$-improving and Sobolev smoothing estimates) for "generic" degenerate averaging operators.