Heavy particle expansions, familiar from heavy quark physics, have found important applications in the analysis of dark matter candidates and their interactions with the Standard Model. From a different direction, preciion spectroscopy of muonic hydrogen has challenged QED and required more precise knowledge of proton structure. These problems have forced a closer examination of the construction of general heavy particle lagrangians at high orders in the 1/M expansion, and in the absence of known ultraviolet completions. Key aspects of this formalism, including the emergence of Lorentz invariance from "nonrelativistic" lagrangians, are reviewd, and several applicatins are presented. A status report on the proton radius puzzle is given.
