Computational fluid dynamics (CFD) codes today represent consolidated tools that cover most physical and chemical processes which occur during operation of internal combustion engines under steady and unsteady conditions. Despite the availability of advanced physical models, the most demanding prerequisite for a CFD engine code is its flexibility in mesh structure and geometry handling capacity to accommodate moving boundaries. In fact, while the motion is solely defined on boundary points, most CFD approaches a-priori specify the position of every mesh vertex in the mesh for every time-step. Alternatives exist, most commonly using mesh generation techniques, like smoothing. In practice this is quite limiting, as it becomes difficult to prescribe solution-dependent motion or perform mesh motion on dynamically adapting meshes. To preserve the mesh quality during extreme boundary deformation due to piston and valve motion, the number of the cells in the mesh needs to be changed. For this reason a set of “topological changes” has been defined, allowing the possibility of attaching or detaching boundaries, adding or removing cell layers and using sliding mesh interfaces. This paper presents a new method of handling motion and topological changes in a moving-mesh Finite Volume Method (FVM). It consists of self-contained mesh topology modifiers which operate without direct “event” prescription and a vertex-based unstructured mesh motion solver. Mesh motion is determined by solving a motion equation with variable diffusion on mesh points, using a Finite Element method with polyhedral cell support. This guarantees that an initial valid mesh remains geometrically valid for arbitrary boundary motion. The proposed algorithms have been implemented in the OpenFOAM code, in order to perform mesh motion on the most common engine geometries. The proposed mesh setup relies on a static initial mesh which can be generated by standard commercial and open-source tools. The final part of the work concerns the application of the Finite Volume Method on moving meshes with topological changes, considering different test cases and preliminary results of the simulation of in-cylinder gas-exchange phase and combustion process for simplified geometries.