An interesting phenomenon, later on termed solid flame, was discovered in 1967 by the author in collaboration with I. P. Borovinskaya and V. M. Shkiro. At that time, it was considered as an unconventional burning mode of solids yielding only solid reaction products with no gas or liquid. In the solid flames, the predominant transport processes are heat conduction and diffusion. Studies of solid flames resulted in the development of the method of self-propagating high-temperature synthesis (SHS) of inorganic compounds and materials. The method has found wide practical applications and substantially extended the scope of scientific enquiry. Studies then were initiated on the processes of the solid flame type involving liquids and gases as reactants. Attention was paid to a wide class of reactions, in which solid products were formed via liquid or gaseous intermediates. In these processes, of importance are the more complicated modes of mass transfer, such as convection and filtration. Novel phenomena of the fluid dynamics nature have been found to occur in SHS waves. This paper is the first attempt to thoroughly consider and systematize the fluid dynamics phenomena taking place in the SHS waves of various types, such as gasless burning and the filtration- and condensation-controlled SHS processes. Also discussed are the phenomena of impurity degassing and self-purification, capillary spreading, gravitational impregnation, convective phase separation in multicomponent melts, surface and layer-by-layer burning, bifurcation and reflection of combustion fronts, filtration extinction, superadiabatic temperature rise, etc