This book describes the research done on the problems of Benard convection, its modern offspring the RayleighSHBenard problem, and Taylor vortices. Both problems are classical examples in the theory of hydrodynamic stability and share many features. While linear theory describing the onset of instability for both problems is practically completed, nonlinear problems are at the forefront of research. Thus the author concentrates on the impressive progress that has been made in the theoretical and experimental investigation of the nonlinear problems and outlines the remaining basic problems.