Wakes of bluff bodies have long been considered a major area of interest within fluid dynamics due to their effect on drag production, pressure recovery, forces, vortex shedding, flow-induced vibrations, and wake mixing. Circular and square cylinders represent some canonical cases of bluff bodies that possess nearly similar features when it comes to the wake behavior; however, they significantly vary from each other concerning the separation process, growth of the near wake, and instabilities within the flow field. This paper offers an analysis of the evolution of wakes generated by both circular and square cylinders, focusing on the discussion of the mathematical and physical aspects related to the formation, development, and instability associated with the wake, vortex shedding, and the resulting force variation. Equations of incompressible flows are provided, while the evolution of the studied cases is analyzed via nondimensional numbers, including Reynolds number, Strouhal number, drag coefficient, and lift coefficient. Further mathematical analysis is performed via wall-stress-based separation criteria and perturbation-based concepts of instabilities. The findings show that for the circular cylinder, separation occurs through boundary-layer development and pressure gradients, while for the square cylinder, separation occurs through corners, which produce a wider wake and pressure deficits in the base region. With increasing Reynolds number, the flows on the two geometries evolve from steady, symmetric wakes to periodic, unsteady vortices, but the manner in which the flows do so is different owing to their shapes. It is concluded in the study that flow evolution in wakes depends upon inertia, viscosity, and geometric sharpness of the body shape, all of which need to be characterized mathematically.