Nowadays, machine learning (ML) methods rapidly evolve for their use in model-based control applications. Model-based control requires an accurate model description of the dynamical system to reassure the performance of the controller. Conventionally, this model description is retrieved from first-principles modelling which can be problematic if the system consists of high-order and/or time-varying dynamics. In these applications, ML methods may benefit because of their promising potential to model complicated system behaviour from data. In the field of ML methods for the modelling and control of dynamical systems, Gaussian processes (GPs) form an interesting opportunity. The ability of GPs to directly learn nonlinear system dynamics prevents huge costs and/or efforts when modelling complex systems. GP dynamical models are capable of accurately predicting the behaviour of dynamical systems while also measuring the confidence level of the prediction. For making predictions, GPs do not require huge amounts of data which benefits them over other ML methods. The powerful model predictive control (MPC) and the fully data-driven dynamical modelling capabilities of GPs makes their combination an interesting candidate for sophisticated control systems. MPC has advantages over other control methods since the controller allows operational constraints that provide freedom in controller design or prevent the system to be steered in an infeasible direction, and, the controller is easily extendable to nonlinear and multivariable control. Next to this, an MPC controller is naturally modified to incorporate GP dynamical models (GP-MPC). The GP-MPC controller exploits the GP dynamical model for making predictions over the prediction horizon while it is also possible to incorporate the confidence of the predictions for increased robustness of the controller. Whereas GP-MPC is studied extensively as an augmented model for other modelling techniques, fully data-driven GP-MPC approaches are also deemed to be promising. This thesis considers the use of GPs for learning and modelling dynamic systems for incorporation in a realtime MPC application. The dynamic system studied in this thesis is a double pendulum system in both a simulation and a real-world environment. The simulation provides initial insights into the problem and allows the rapid development of an algorithm. Eventually, the proposed realtime GP-MPC algorithm is tested on a physical laboratory-scale setup. The results show that the realtime data-driven controller is able to track a reference with high accuracy, while also being robust to disturbances. These promising results on GPs for realtime nonlinear control might be a step for GPs to be incorporated into future control systems.