High-gain free-electron lasers (FELs) are being developed as extremely bright sources for a next-generation x-ray facility. In this paper, we review the basic theory of the start-up, the exponential growth, and the saturation of the high-gain process, emphasizing the self-amplified spontaneous emission. The radiation characteristics of an x-ray FEL, including its transverse coherence, temporal characteristics, and harmonic content, are discussed. FEL performance in the presence of machine errors and undulator wakefields is examined. Various enhancement schemes through seeding and beam manipulations are summarized.