This paper discusses a multilinear generalization of the singular value decomposition (SVD) for higher-order tensors. It explores the analogy between matrix and tensor properties, such as uniqueness, link with matrix eigenvalue decomposition, and first-order perturbation effects. The paper investigates how tensor symmetries affect the decomposition and proposes a multilinear generalization of the symmetric eigenvalue decomposition for pair-wise symmetric tensors. The authors derive the tensor decomposition in an SVD terminology, using a notation that extends the notation used for matrices. They show that the Tucker model, originally developed in psychometrics, is a convincing multilinear generalization of the SVD concept. The paper also discusses the rank properties of higher-order tensors, the scalar product, orthogonality, and norm of higher-order tensors, and the multiplication of a higher-order tensor by a matrix. The authors provide a detailed derivation of the multilinear SVD and demonstrate its connection to the matrix SVD through matrix unfoldings. They prove that the HOSVD of a tensor can be computed by finding the left singular matrix and singular values of an $n$-mode matrix unfolding of the tensor. The paper concludes with a discussion of various properties of the multilinear SVD, including uniqueness, generalization, rank, structure, norm, oriented energy, and approximation.This paper discusses a multilinear generalization of the singular value decomposition (SVD) for higher-order tensors. It explores the analogy between matrix and tensor properties, such as uniqueness, link with matrix eigenvalue decomposition, and first-order perturbation effects. The paper investigates how tensor symmetries affect the decomposition and proposes a multilinear generalization of the symmetric eigenvalue decomposition for pair-wise symmetric tensors. The authors derive the tensor decomposition in an SVD terminology, using a notation that extends the notation used for matrices. They show that the Tucker model, originally developed in psychometrics, is a convincing multilinear generalization of the SVD concept. The paper also discusses the rank properties of higher-order tensors, the scalar product, orthogonality, and norm of higher-order tensors, and the multiplication of a higher-order tensor by a matrix. The authors provide a detailed derivation of the multilinear SVD and demonstrate its connection to the matrix SVD through matrix unfoldings. They prove that the HOSVD of a tensor can be computed by finding the left singular matrix and singular values of an $n$-mode matrix unfolding of the tensor. The paper concludes with a discussion of various properties of the multilinear SVD, including uniqueness, generalization, rank, structure, norm, oriented energy, and approximation.