We are interested in the fast solution of nonlinear ODE/DAE-constrained mixed-integer optimal control and model predictive control problems. Such problems frequently arise in industrial process control, and typically show significant potential for optimization. The hybrid and nonlinear nature of these problems however is challenging to deal with. We present a computational framework based on a direct and simultaneous method for optimal control and on a partial outer convexification reformulation of the problem. We show how to efficiently compute approximate solutions with feasibility and optimality certificates, and can typically do so without experiencing exponential runtime. The concept of real-time iterations also allows for a transfer of our framework to closed-loop control. Here, the computational performance is determined by the effort required to solve one nonconvex feedback QP in each real-time iteration. Block structures are exploited to significantly reduce this effort. We conclude with an outlook on current algorithmic developments in mixed-integer nonlinear model-predictive control.