This paper reviews methods for designing and learning kernel functions for multiple outputs, focusing on the connection between probabilistic and functional approaches. It begins by discussing the classical problem of learning scalar outputs using kernel methods, including both regularization and Bayesian perspectives. The paper then extends these concepts to vector-valued functions, describing the reproducing kernel Hilbert space (RKHS) and Gaussian process (GP) frameworks for multi-output learning. The authors explore separable kernels and sum of separable kernels (SoS kernels), which can be formulated as sums of products between a kernel function for the input space and a kernel encoding interactions among outputs. They also discuss coregionalization models, such as the linear model of coregionalization (LMC) and the intrinsic coregionalization model (ICM), which are useful for developing valid covariance functions. The paper concludes with a discussion on the computational aspects and potential applications of multi-output kernels.This paper reviews methods for designing and learning kernel functions for multiple outputs, focusing on the connection between probabilistic and functional approaches. It begins by discussing the classical problem of learning scalar outputs using kernel methods, including both regularization and Bayesian perspectives. The paper then extends these concepts to vector-valued functions, describing the reproducing kernel Hilbert space (RKHS) and Gaussian process (GP) frameworks for multi-output learning. The authors explore separable kernels and sum of separable kernels (SoS kernels), which can be formulated as sums of products between a kernel function for the input space and a kernel encoding interactions among outputs. They also discuss coregionalization models, such as the linear model of coregionalization (LMC) and the intrinsic coregionalization model (ICM), which are useful for developing valid covariance functions. The paper concludes with a discussion on the computational aspects and potential applications of multi-output kernels.