The problem of heat transfer by convection in a horizontal layer of fluid across which the imposed temperature gradient gives rise to an unstable density gradient is classical in the theory of fluid mechanics. The central question, of the instability to small perturbations of a layer of incompressible, non-radiating fluid contained between rigid horizontal boundaries, has been completely answered, and has provided the key to a number of investigations of the patterns of finite amplitude convection that do indeed occur in unstable conditions. In recent years an increased awareness of the importance of convective processes in many situations of geophysical and astrophysical interest has stimulated a considerable amount of theoretical work on convective instability and heat transport under conditions rather less restrictive than those of the central problem mentioned. For example, the effects of radiative heat transfer and compressibility have been taken into account, as has the effect of conditional instability in meteorology, and the penetration of convective motions into possibly stable bounding layers when the conditions of horizontal rigid boundaries are relaxed. The purpose of the present paper is to review and criticize these recent developments, and, it is hoped, to focus attention on the areas of greatest theoretical inadequacy. It is not intended that the paper should be merely a bibliography, nor does it claim to be complete in this respect, but rather an exposition of the present (early 1966) state of the theory. Consequently, certain papers, not necessarily the first or best in their field but those which in the writer's opinion expose important features of the problem, are widely quoted, whilst other similar papers may be only briefly acknowledged.