While model predictive control (MPC) methods have proven their efficacy when applied to systems with safety specifications and physical limitations, their performance heavily relies on an accurate prediction model. As a consequence, a significant effort in the design of MPC controllers is dedicated to the modeling part and often requires advanced physical expertise. In order to facilitate the controller design, we present an MPC scheme supporting nonlinear learning-based prediction models, that is, data-driven models with probabilistic parameter uncertainties. A tube-based MPC formulation in combination with an additional implicit state and input constraint forces the closed-loop system to be operated in domains of sufficient model confidence, thereby ensuring asymptotic stability and constraint satisfaction at a prespecified level of probability. Furthermore, by relying on tube-based MPC concepts, the proposed learning-based MPC formulation offers a general framework for addressing different problem classes, such as economic MPC, while providing a general interface to probabilistic prediction models based, for example, on Bayesian regression or Gaussian processes. A design procedure is proposed for approximately linear systems and the efficiency of the method is illustrated using numerical examples.