Although it has been a subject of research for years (e.g. biophysict Victor Small) mechanisms allowing living cells to move around the body are not completely understood. These differ between cell type and a given cell can be sport several of them. For example leukocytes can move on a surface by sticking to it at several locations (focal adhesion) and rolling forward, similarly as bulldozer tracks. Concerning this particular mechanism, recent experiments at the IST (Reversat & Sixt), leukocytes were engineered and stripped from their adhesion capabilities. When placed in appropriately structured media, these cells are still able to move around the environment. I will discuss the experiments and two distinct variants of a new mechanical model describing this behavior, based on simple physical considerations. The two key ingredients that we consider are the renewal of the actin cortex through polymerization and cortex internal viscosity, which when combined, create motion. The resulting system of parabolic equations is of integro-differential type and involves high-order in space differential operators. It can be analyzed partially, and existence results will be given in simple situations. I will also discuss some numerical experiments and an extension of the model which includes the mechanical contribution of the cell's nucleus. This is a joint work with C. Schmeiser, D. Peurichard, L. Preziosi and C. Giverso