Smoothed Particle Hydrodynamics (SPH) is a numerical method that solves the Lagrangian, conservative equations of mass, momentum, and energy without using a computational grid. In the past, SPH has been used to solve problems of astrophysics, hypervelocity collisions, and explosions. This work has extended the application of SPH by solving problems involving inviscid, compressible aerodynamic flow. New boundary conditions were developed, so SPH could solve these types of problems. This paper will provide a short history of SPH, an introduction to the method, discuss the transformation of the governing fluid equations into the SPH formulation, and present several aerodynamic test cases. The power of SPH is that it requires no computational grid. This feature may significantly reduce the time required for the engineer to calculate computational solutions or may simplify the numerical techniques used to solve complex problems, such as moving body problems.