Mesh-free Lagrangian (Particle) Methods are the latest generation of numerical methods in the area Computational Fluid Dynamics (CFD) and are known to be capable of dealing with large deformations and fragmentation of boundaries and interfaces. The Moving Particle Semi-implicit (MPS) method is one these methods, widely used for fluid mechanics applications. The present monograph (the author's Ph.D. dissertation) provides a comprehensive review of the MPS method and its mathematical fundamentals. It also presents the author's contributions in developing of the Weakly Compressible MPS(WC-MPS) method, the multiphase MPS method and extending the application of MPS for open-channel flow and granular flow. The sample problems used in this work are the fundamental and practical free-surface and interfacial flow problems such as the dam-break flow, flow in open-channels, stratified flow and sediment transport.