We propose a model predictive control approach for non-linear systems based on linear parameter-varying representations. The non-linear dynamics are assumed to be embedded inside an LPV representation. Hence, the non-linear MPC problem is replaced by an LPV MPC problem, which can be solved through convex optimization. Doing so, the non-linear system can be controlled efficiently and with strong guarantees on feasibility and stability at the possible sacrifice of achievable performance. In this paper, the LPV MPC problem is solved using a tube-based approach, requiring the on-line solution of a single linear-or quadratic program. The computational properties of the approach are demonstrated on two examples.