In an attempt to introduce application scientists and graduate students to the exciting topic of positive definite kernels and radial basis functions, this book presents modern theoretical results on kernel-based approximation methods and demonstrates their implementation in a variety of fields of application. With the aim of providing researchers involved in function approximation, boundary value problems, spatial statistics and machine learning with the flexible and high-order tools developed using kernels, the authors explore their historical context and explain recent advances as strategies to address long-standing problems.The examples are drawn from fields as diverse as surrogate modeling, machine learning and finance, and researchers from those and other fields will be able to follow the examples on their own machines using the included MATLAB code accessible through the library online.In combining the theoretical foundation of positive definite kernels with accessible experimentation from which to build on, the authors are empowering readers to use these powerful tools on their problems of interest.