Utilizing a crossdisciplinary approach, we explore Shannon information-theoretic characterizations of the information capacity limits of generic electromagnetic surfaces intended for possible use in wireless communication links. Our principal task is to first formulate at a general and rigorous level the electromagnetic theory of the information that can be extracted from the Maxwellian fields radiated by an arbitrarily-shaped continuous surface. This is then followed by a detailed derivation and illustration of practical physics-informed algorithms for computing approximations of the Shannon capacity of surfaces with any given geometry operating in Gaussian (AWGN) channels. Our formalism can address both near- and far-field information capacity scenarios, with a mathematical treatment that includes a complete characterization of the source-field polarization structure, mutual coupling, and interactions.