A Free-Electron Laser (FEL) operating in the quantum regime can provide a compact and monochromatic X-ray source. Here we review the basic principles of a high-gain quantum FEL starting from noise, with special emphasis on the self-amplified spontaneous emission (SASE) mode operation. In the first part, the full quantum theory of the N-particle and single-radiation-mode FEL Hamiltonian is presented. Quantum effects such as cooperative gain, discrete spectrum and line narrowing are described, both in the multi-particle and in the second quantization formalism. In the second part, propagation effects (i.e. slippage) are described and the main features of the quantum SASE regime are discussed. The broad and spiky radiation spectrum observed in the classical SASE reduces in the quantum regime to a series of narrow lines, associated to sequential transitions between adjacent momentum states. A simple interpretation of the discrete nature of the spectrum and of the line width of the single spike observed in the quantum regime is presented.