We will investigate the regime of low and high Mach number flows for compressible fluids. The Mach number, physically, is given by the ratio of the reference fluid velocity and the sound speed. In general the value of the Mach number depends on the conditions of the physical phenomenon under consideration and it is related to the compressibility of the fluid according to its low or high values. In the case of a low Mach number regime the speed of sound tends to infinity while the pressure becomes almost constant and doesn’t generates density variations, as a consequence compressibility can be ignored and the final asymptotical physical state is the incompressible one. Conversely if the fluid speed increases beyond the sound speed then the Mach number is high and the compressibility effects have to be taken into consideration. We will discuss through different examples how to perform rigorously these limits in the two different regimes and what are the main differences and difficulties from a mathematical point of view.