This book presents an overview of reproducing kernel Hilbert, Sobolev, and Holder spaces. Unlike most related books, this one brings together the two perspectives of geostatistics and reproducing kernel Hilbert spaces. It begins with second-order random fields defined over Hilbert spaces and then progresses to applications to space, space-time random fields, vector-valued random fields, functional geostatistics, and extreme-valued random fields. The book includes real-world examples and case studies that illustrate data analysis and simulation. All examples are implemented with R packages.