A theory of photoemission is presented in which all results are derived rigorously from first principles. It is shown how to calculate properly the external current of electrons, with the transmission at the surface done correctly. A new and rigorous formalism is derived for doing many-body calculations in photoemission. Extensive calculations are performed on the angular dependence of photoemission. It is shown that the angular anisotropy is interesting and significant. Numerous numerical examples are presented for the alkali metals.