We first introduce Gaussian Process (GP) emulation of computer models. These are surrogates of simulators that efficiently mimic the input-output relationship of such complex numerical models, only sampling a small set of runs. GPs crucially model uncertainties. We illustrate by embedding emulation of convection within a climate numerical model, with gains of 20% in reducing precipitation biases. We then present a new type of emulator of any feed forward multi-physics system, by linking GP emulators of individual simulators, with large gains over the composite emulator of the whole system. The Deep Gaussian Process (DGP) is then presented as a surrogate that shares the structure of the linked emulator but enables the emulation of highly non-linear simulators without the knowledge of individual sub-processes. We then examine sharp changes in the outputs a computer simulator. These often indicate bifurcations or critical transitions within the investigated system, e.g. laminar v. turbulent behavior in fluid dynamics. An efficient approach that localizes these changes using DGPs with minimal number of evaluations is introduced. We demonstrate the efficacy of the proposed framework on the Rayleigh-Bénard convection. Finally we present recent results in solving inverse problems by expanding the history matching technique, with applications to the fast detection with buoys and satellites of the 2004 Indian Ocean tsunami and the 2011 Tōhoku tsunami.