This text aims to provide a comprehensive study of the mathematical theory of the vortex method, from its origins in the 1930s, through the developments of the 70s when the use of computers made advanced research possible, to current work on this subject in China and elsewhere. The five chapters treat vortex methods for the Euler and Navier-Stokes equations; mathematical theory for incompressible flows; convergence of vortex methods for the Euler equations; convergence of viscosity splitting; and convergence of the random vortex method.