In this thesis, several challenges in both ground-motion modelling and the surrogate modelling, are addressed by developing methods based on Gaussian processes (GP). The first chapter contains an overview of the GP and summarises the key findings of the rest of the thesis. In the second chapter, an estimation algorithm, called the Scoring estimation approach, is developed to train GP-based ground-motion models with spatial correlation. The Scoring estimation approach is introduced theoretically and numerically, and it is proven to have desirable properties on convergence and computation. It is a statistically robust method, producing consistent and statistically efficient estimators of spatial correlation parameters. The predictive performance of the estimated ground-motion model is assessed by a simulation-based application, which gives important implications on the seismic risk assessment. In the third chapter, a GP-based surrogate model, called the integrated emulator, is introduced to emulate a system of multiple computer models. It generalises the state-of-the-art linked emulator for a system of two computer models and considers a variety of kernels (exponential, squared exponential, and two key Matérn kernels) that are essential in advanced applications. By learning the system structure, the integrated emulator outperforms the composite emulator, which emulates the entire system using only global inputs and outputs. Furthermore, its analytic expressions allow a fast and efficient design algorithm that could yield significant computational and predictive gains by allocating different runs to individual computer models based on their heterogeneous functional complexity. The benefits of the integrated emulator are demonstrated in a series of synthetic experiments and a feed-back coupled fire-detection satellite model. Finally, the developed method underlying the integrated emulator is used to construct a non-stationary Gaussian process model based on deep Gaussian hierarchy
