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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The moving particle semi-implicit (MPS) method is a computational method for the simulation of incompressible free surface flows. It is a macroscopic, deterministic particle method (Lagrangian meshfree method) developed by Koshizuka and Oka (1996).The MPS method is similar to the SPH (smoothed-particle hydrodynamics) method (Gingold and Monaghan, 1977; Lucy, 1977) in that both methods provide approximations to the strong form of the Partial Differential Equations (PDEs) on the basis of integral interpolants. However, the MPS method applies simplified differential operator models solely based on a local weighted averaging process without taking the gradient of a kernel function. In addition, the solution process of MPS method differs to that of the original SPH method as the solutions to the PDEs are obtained through a semi-implicit prediction-correction process rather than the fully explicit one in original SPH method.