The appearance of secondary motions in a viscous fluid field can be understood to some extent as a bifurcation phenomenon with exchange of stability between the basic and the secondary flow. This article summarizes the main mathematical results of bifurcation and stability in hydrodynamic stability theory so far obtained. A unified functional-analytic approach is presented which tries to accentuate the ideas and to avoid technicalities. Besides the general results on the existence, the number of solutions and their qualitative behavior, the constructive analytical methods are emphasized. The Taylor and the Benard models are studied in detail. In the latter case, all possible solutions of regular cell pattern are classified. Stability and instability and their exchange at the point of bifurcation are studied.