A review is presented of the types of instabilities which are encountered in pumping systems of technological interest. These include axial and centrifugal compression systems, pumping systems involving cavitation, systems with two-phase flow, systems with combustion, hydraulic systems, and systems which have two or more pumping elements in parallel. All of the above will be seen to exhibit instabilities under certain operating conditions, although the mechanism of instability, as well as the particular system element that is responsible for the instability, will be quite different in the different systems. However, several basic concepts, such as the idea of negative damping which is associated with dynamic instability, will be seen to be common to the different systems. The review is organized around the different types of systems that are discussed, and includes descriptions of the steady-state performance, the regimes in which one would expect instability, and the mechanisms of instability. An idealized pumping system is first examined to illustrate some of the basic concepts. More realistic systems are then treated in the same manner of showing steady-state performance, regimes of instability and mechanisms. In the review attention is given mainly to those areas in which there is high current engineering interest, and an attempt is made to describe those areas of research which can be most fruitfully pursued. In general, it is suggested that efforts should be directed toward obtaining an improved understanding of the transient behavior of the active (instability causing) elements within the system, since it is lack of knowledge of this aspect that currently limits the accuracy of system stability predictions.