This book describes the research done on the problems of Bnard convection, as well as its modern offspring the Rayleigh-Bnard problem, and Taylor vortices. Bnard convection differs from Rayleigh-Bnard convection by the presence of surface tension, whilst Bnard convection is characterized by parallel rolls. Toroidal vortices characterize Taylor vortex flow. Convection and Taylor vortex flow deal with the consequences of the presence of infinitesimal disturbances in a fluid layer. Both problems are classical examples in the theory of hydrodynamic stability and share many features. Linear theory describing the onset of instability for both problems is practically completed; nonlinear problems have been at the forefront of research during the last 30 years. The impressive progress that has been made in the theoretical and experimental investigation of the nonlinear problems is described and the remaining basic problems are outlined.