Aim of the present chapter is to provide an in-depth presentation and survey of mesh-free particle methods. Several particle approximation are reviewed; the SPH method, corrected gradient methods and the Moving least squares (MLS) approximation. The discrete equations are derived from a collocation scheme or as Galerkin method. Special attention is paid to the treatment of essential boundary conditions. A brief review over radial basis functions is given because they play a significant role in mesh-free methods. Finally, different approaches for modelling discontinuities in mesh-free methods are described.