The paper reviews machine learning methods that use positive definite kernels. These methods formulate learning and estimation problems in a reproducing kernel Hilbert space (RKHS) of functions defined on the data domain. Working in linear spaces of functions allows for the construction and analysis of learning algorithms while enabling a wide range of functions, including nonlinear ones. The paper covers a range of methods, from binary classifiers to sophisticated methods for estimation with structured data. The paper discusses the properties of kernels, focusing on positive definite kernels and their characterization. It provides examples of such kernels and discusses kernels and RKHS in the context of regularization. It also presents various approaches for estimating dependencies and analyzing data using kernels, including an overview of problem formulations and their solutions using convex programming techniques. Finally, it examines the use of RKHS as a means to define statistical models, focusing on structured, multidimensional responses and how such techniques can be combined with Markov networks. The paper also discusses the construction of the reproducing kernel Hilbert space, properties of positive definite kernels, and examples of kernels such as polynomial, spline, ANOVA, and string kernels. It highlights the importance of positive definite kernels in machine learning and their ability to capture complex relationships in data. The paper concludes with a discussion of kernel function classes, the representer theorem, and the regularization properties of kernels in the Fourier domain.The paper reviews machine learning methods that use positive definite kernels. These methods formulate learning and estimation problems in a reproducing kernel Hilbert space (RKHS) of functions defined on the data domain. Working in linear spaces of functions allows for the construction and analysis of learning algorithms while enabling a wide range of functions, including nonlinear ones. The paper covers a range of methods, from binary classifiers to sophisticated methods for estimation with structured data. The paper discusses the properties of kernels, focusing on positive definite kernels and their characterization. It provides examples of such kernels and discusses kernels and RKHS in the context of regularization. It also presents various approaches for estimating dependencies and analyzing data using kernels, including an overview of problem formulations and their solutions using convex programming techniques. Finally, it examines the use of RKHS as a means to define statistical models, focusing on structured, multidimensional responses and how such techniques can be combined with Markov networks. The paper also discusses the construction of the reproducing kernel Hilbert space, properties of positive definite kernels, and examples of kernels such as polynomial, spline, ANOVA, and string kernels. It highlights the importance of positive definite kernels in machine learning and their ability to capture complex relationships in data. The paper concludes with a discussion of kernel function classes, the representer theorem, and the regularization properties of kernels in the Fourier domain.