This article contains a review of combustion instabilities coupled by pressure waves and of the related pulse combustor and active control principles and applications. The basic mechanisms leading to combustion oscillations are first identified from experimental evidence gathered in recent years. It is shown that in the presence of pressure waves, the flow features large scale motions which drive the instability. The dynamics of the flame is then dominated by processes of hydrodynamic instability, vortex roll-up, vortex interactions, front and reacting stream pulsations, periodic extinctions and reignitions, self acceleration. Knowledge accumulated on these various aspects has provided guidelines in combustor design. Progress accomplished in the special domain of pulse combustors is surveyed. These devices which operate under sustained pressure oscillations have been the subject of intensive research with as objective the understanding of the dynamics of the flow in various geometrical situations in order to improve the heat transfer efficiency and diminish the level of pollutant formation. While pulse combustors use pressure oscillations in normal operation, in most circumstances the presence of pressure waves causes problems of vibration, enhanced heat fluxes to the walls, cyclic fatigue, and in some cases important damage to the combustor. One generally wishes to eliminate or attenuate these waves. This is usually achieved by changes in the geometrical design and in the operation of the system. New possibilities based on active control have been investigated in recent years. The basic principles and the initial demonstrations of active instability control are first reviewed. It is then shown that such methods constitute a promising technology and that they are also a unique tool in the experimental investigation of combustion instability and pressure related phenomena.