Information theory establishes the ultimate limits on performance for noisy communication systems(1). Accurate models of physical communication devices must include quantum effects, but these typically make the theory intractable(2-5). As a result, communication capacities-the maximum possible rates of data transmission-are not known, even for transmission between two users connected by an electromagnetic waveguide with Gaussian noise(6). Here we present an exactly solvable model of communication with a fully quantum electromagnetic field. This gives explicit expressions for all point-to-point capacities of noisy quantum channels, with implications for quantum key distribution and fibre-optic communications. We also develop a theory of quantum communication networks by solving some rudimentary models including broadcast and multiple-access channels. We compare the predictions of our model with the orthodox Gaussian model and in all cases find agreement to within a few bits. At high signal-to-noise ratios, our simple model captures the relevant physics while remaining amenable to exact solution.