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In this paper, a framework for nonlinear model predictive control (NMPC) for heavily noise-affected systems is presented. Within this framework, the noise influence, which originates from uncertainties during model identification or measurement, is explicitly considered. This leads to a significant increase in the control quality. One part of the proposed framework is the efficient state prediction, which is necessary for NMPC. It is based on transition density approximation by hybrid transition densities, which allows efficient closed-form state prediction of time-variant nonlinear systems with continuous state spaces in discrete time. Another part of the framework is a versatile value function representation using Gaussian mixtures, Dirac mixtures, and even a combination of both. Based on these methods, an efficient closed-form algorithm for calculating an approximate value function of the NMPC optimal control problem employing dynamic programming is presented. Thus, also very long optimization horizons can be used and furthermore it is possible to calculate the value function off-line, which reduces the on-line computational burden significantly. The capabilities of the framework and especially the benefits that can be gained by incorporating the noise in the controller are illustrated by the example of a miniature walking robot following a given path.