We review the theory and use of phase retrieval and phase diversity in astronomy. Phase retrieval (PR) uses the image of a star to estimate T, the phase aberration introduced by the atmosphere and the telescope. Phase diversity (PD) is an extension of PR in which changes (diversities) are added to T so that both T and an extended object under observation can be estimated. In 1990, when the Hubble Space Telescope was found to have an optical flaw both PD and PR were used to help scientists determine a prescription to fix the flaw. A more recent use includes fine phasing of the test bed for the James Webb Space Telescope. Proposed uses include exoplanet imaging and sequential diversity imaging, in which sequential changes in the AO are the diversities. The major advantage of these methods is that they need no auxiliary hardware, like a guide star or a Shack-Hartmann wavefront sensor.