We propose virtual element methods for solving one- or multi-domain elliptic interface problems using interface-fitted meshes. The main challenge is to design an efficient and robust mesh that can capture certain properties while preserving arbitrary complex geometries of the interface. Moreover, it is a tricky problem in classical finite element methods when the domain is decomposed into tetrahedra due to the existence of slivers in three dimensions. In our mesh generation, every element in three dimensions could be any polyhedron instead of tetrahedron. Then we apply virtual element methods for solving elliptic interface problems with solution and flux jump conditions. The purpose of using virtual element methods rather than classical finite element methods is that every element could be a different shape. We use multi-grid solvers to solve the discrete system. Lastly, numerical examples demonstrating the theoretical results for linear elements are shown.