Abstract The current state of the classical theory of hydrodynamic instability is examined by viewing the mathematical theory (as well as experimental data) concerning the randomization of motions of liquids and gases as a problem in bifurcation theory of families of dynamic systems. Along with a discussion of the theory of linear operators encountered in hydrodynamics (a theory which is still not entirely complete), the author also gives illustrations of powerful nonlinear methods used in the analysis of hydrodynamic instability, such as Landau's amplitude equations and V. I. Arnold's variational method. The multiplicity of possible scenarios for randomization of fluid motions is noted, of which the most thoroughly investigated is M. Feigenbaum's universal sequence of period-doubling bifurcations. Recent experimental data concerning the bifurcations of G. Taylor flow between rotating cylinders and E. Lorentz flow in the case of convection in a planar fluid layer are analyzed.