What happens when two masses scatter?  This deceptively simple question has produced a series of surprises that touch on deep issues in general relativity, such as the localizability of gravitational field energy, the notion of asymptotic flatness at timelike infinity, and observability of particle position.  Most recently, we found that the mechanical center of mass of the system undergoes a shift during the scattering process, akin to a person scooting forward on the floor.  This “gravitational scoot” appears at the first post-Minkowskian order (1PM) in the traditional order-counting, but appears to be essentially non-perturbative in nature.  It also occurs in electromagnetism, where it can be clearly understood as a permanent exchange of center-of-mass between particles and field.  In this talk I will review recent developments in gravitational scattering, discuss our novel calculation of 2PM scattering using self-force methods, and present the gravitational and electromagnetic scoots.