This course provides a thorough and comprehensive exploration of Gaussian processes and kernel methods, bridging foundational theory with practical applications in regression, learning, and numerical analysis. It covers Gaussian vectors, processes, fields, and measures, with a particular focus on regression techniques. The course delves into kernel methods and Reproducing Kernel Hilbert Spaces (RKHS), examining key concepts such as Kernel PCA, LDA, CCA, kernel mean embedding, and operator-valued kernels. A central theme will be the interplay between kernel methods and optimal recovery techniques, with applications in statistical numerical approximation, signal processing, and machine learning.