This thesis presents extensions of the Gaussian Process (GP) model, based on approximate methods allowing the model to deal with input uncertainty. Zero-mean GPs with Gaussian covariance function are of particular interest, as they allow to carry out many derivations exactly, as well as having been shown to have modelling abilities and predictive performance comparable to that of neural networks (Rasmussen, 1996a). With this model, given observed data and a new input, making a prediction corresponds to computing the (Gaussian) predictive distribution of the associated output, whose mean can be used as an estimate. This way, the predictive variance provides error-bars or confidence intervals on this estimate: It quantifies the model's degree of belief in its 'best guess'. Using the knowledge of the predictive variance in an informative manner is at the centre of this thesis, as the problems of how to propagate it in the model, how to account for it when derivative observations are available, and how to derive a control law with a cautious behaviour are addressed. The task of making a prediction when the new input presented to the model is noisy is introduced. Assuming a normally distributed input, only the mean and variance of the corresponding non-Gaussian predictive distribution are computed (Gaussian approximation). Depending on the parametric form of the covariance function of the process, exact or approximate moments are derived. These results are then used for the multiple-step-ahead iterative forecasting of nonlinear dynamic systems, with propagation of the uncertainty. Within a nonlinear auto-regressive representation of the system, modelled by a GP, a one-step-ahead model is iterated up to the desired horizon. At each time-step, the uncertainty induced by each successive prediction is propagated, that is, the whole predictive distribution of the output just predicted is fed back into the state for the next time-step. Not only are the predictive variances of each delayed output accounted for, but also the cross-covariances between them. The approach is illustrated on the simulated Mackey-Glass chaotic time-series, as well as on two real-life dynamic processes, a gas-liquid separator and a pH neutralisation process. The emphasis is on the use of Gaussian Processes for modelling nonlinear dynamic systems. GPs have not yet gained widespread popularity among the engineering community. It is well known that the modelling of such systems is in practice rendered difficult by the fact that most available data lies around equilibrium regions, and very few points in transient areas, and a common approach has been to consider linearisations around those equilibrium points. Derivative observations can be elegantly integrated in a GP model where function observations are already available, as shown in (Solak et al., 2003). As well as being in accord with engineering practice, derivative observations can potentially reduce the computational burden usually associated with GPs (typically, a linear region can be summarised by one derivative observation, instead of many function observations). For this mixed training set, the explicit expressions of the predictive mean and variance of a function output corresponding to a noise-free and to a noisy input are then derived, the latter being tackled within the Gaussian approximation. The other field where GPs can present an advantage other over models is in the control of nonlinear dynamic systems. Commonly, researchers have used nonlinear parametric models and have adopted the certainty equivalence principle when deriving the control law, whereby the model's predictions are used as if they were the true system's outputs. Deriving controllers with 'cautious' and 'probing' features is difficult and has been the scope of much work in the control community. The propagation of uncertainty method is applied for a cautious controller, where the cautiousness is accounted for in a cost function that does not disregard the variance associated with the model's estimate.