Axions can be described by a relativistic field theory with a real scalar field whose self-interaction potential is a periodic function. If axionic dark matter is a Bose-Einstein condensate, it can be described more simply by a nonrelativistic effective field theory with a complex scalar field. A stable axion star is a gravitationally bound Bose-Einstein condensate of axions. In previously known axion star solutions, the attractive forces of gravity and axion pair interactions are balanced by the kinetic axion pressure. A new class of axion stars has been discovered in which the attractive forces are balanced by the mean-field pressure of the axion Bose-Einstein condensate.