Two-photon emission is a process in which electron transition between quantum levels occurs through the simultaneous emission of two photons. This phenomenon is important for astrophysics and atomic physics, and semiconductor two-photon emission was recently proposed as a compact source of entangled photons, essential for practical quantum information processing, and three orders of magnitude more efficient than the existing down-conversion schemes. Two-photon absorption in semiconductors has been extensively investigated; however, spontaneous semiconductor two-photon emission has not been observed, nor has it been fully analysed theoretically so far. We report the first experimental observations of two-photon emission from semiconductors and develop a corresponding theory. Spontaneous two-photon emission is demonstrated in optically pumped bulk GaAs and in electrically driven GaInP/AlGaInP quantum wells. Singly stimulated two-photon emission measurements demonstrate the theoretically predicted two-photon optical gain in semiconductors 12-15 - a necessary ingredient for any realizations of future two-photon semiconductor lasers. A photon-coincidence experiment is presented to validate the simultaneity of the electrically driven GaInP/AlGaInP two-photon emission, limited only by the detector's temporal resolution.