We address the physics implications of a precision determination of the weak charge of the proton, QWP, from a parity violating elastic electron proton scattering experiment to be performed at the Jefferson Laboratory. We present the Standard Model (SM) expression for QWP including one-loop radiative corrections, and discuss in detail the theoretical uncertainties and missing higher order QCD corrections. Owing to a fortuitous cancellation, the value of QWP is suppressed in the SM, making it a unique place to look for physics beyond the SM. Examples include extra neutral gauge bosons, supersymmetry, and leptoquarks. We argue that a QWP measurement will provide an important complement to both high energy collider experiments and other low energy electroweak measurements. The anticipated experimental precision requires the knowledge of the order alpha_s corrections to the pure electroweak box contributions. We compute these contributions for QWP, as well as for the weak charges of heavy elements as determined from atomic parity violation.

22 pages of LaTeX, 5 figures