We develop particle methods, referred to as Optimal Transportation Meshfree (OTM), for simulating general solid/fluid flows, including fluid-structure interaction and mass advection-diffusion problems. The OTM method combines concepts from Optimal Transportation theory with material-point sampling and max-ent meshfree interpolation. The OTM method generalizes the Benamou-Brenier differential formulation of optimal mass transport to problems including arbitrary geometries and material behavior. The OTM method enforces mass transport and essential boundary conditions exactly and is free from tension instabilities. It is also geometrically exact with respect to advection and volume and it exactly conserves linear and angular momentum. Its convergence characteristics are verified by means standard benchmark problems and weak convergence methods. We illustrate the range and scope of the OTM method by means of selected examples of application.