Gaussian processes provide Bayesian priors over functions and crucial uncertainty quantification tools. They are widely employed in several fields of science and technology, among which geostatistics, computer experiments and machine learning. The GAP project will tackle two major bottlenecks regarding Gaussian processes. (1) Computational developments of Gaussian processes are at a more advanced stage than their corresponding theoretical developments. (2) Gaussian processes are not sufficiently exploited, outside of standard statistics and machine learning tasks, which is a loss of opportunity. These two limitations will be addressed by a team with expertise in mathematical and computational statistics, applied mathematics and machine learning. By cross-fertilising mathematical techniques from asymptotic statistics, probability and functional analysis, as well as methodological and computational developments, the four following axes will be carried-out. Axes (1) and (2) below are more theoretical while Axes (3) and (4) also aim for real data applications, for example in computer experiments, biology, medicine and finance. (1) Posterior concentration rates will be obtained for various deep Gaussian process models. This will provide the first mathematical guarantees for a rich class of Bayesian models, that notably solves the stationarity issues of standard Gaussian process models. (2) Error bounds will be obtained on approximation procedures for large data sets, more precisely inducing points and variational inference. These bounds will tackle theoretically unexplored settings: exact observations and classification. (3) Constrained Gaussian process models will be extended to high dimension, thanks to additive Gaussian processes and variable selection. This will yield the first high-dimensional Gaussian process models guaranteeing to satisfy the constraints on the whole input space. (4) Two novel applications of Gaussian processes will be provided. First, theoretically grounded source separation procedures will be obtained, under fixed-domain asymptotics. Second, Gaussian processes will enable to perform valid statistical inference post selection of regions. On this latter application, a graphical user interface for practitioners will be implemented.