Fluid-particle interaction problems, where solid particles interact with surrounding fluids, play prominent roles in many scientific and engineering fields. In this thesis, we present numerical simulations for the dynamics of a solid particle moving in one or two-phase flows using an extended finite element method combined with a temporary arbitrary Lagrangian-Eulerian technique. In this method, the whole computation including both fluid and solid motion is carried out on a fixed regular mesh. It is efficient compared to conventional approaches that require regeneration or deformation of meshes. The discontinuous characteristics at the particle surface are captured by Heaviside-enriched finite element basis functions and the slip or no-slip boundary conditions at the interfaces are imposed by a penalty method without introducing additional degrees of freedom. In addition, we apply local mesh subdivisions to improve the accuracy near the interface. Temporary mesh movement is considered in the arbitrary Lagrangian-Eulerian formulation of incompressible flows where field variables at the previous time level are mapped onto the computational mesh at the current time level. In numerical experiments, we study the rotation of an ellipsoidal particle in a simple shear flow, the results show good agreement with the Jeffery orbit theory. We also model a particle passing through a fluid-fluid interface and compare the results with the capillary bridge and splashing phenomena observed in experiments.