This paper proposes a theory of eigenvalues, eigenvectors, singular values, and singular vectors for tensors using a constrained variational approach similar to the Rayleigh quotient for symmetric matrices. The approach generalizes the spectral theory of matrices to higher-order tensors. For example, it discusses a multilinear generalization of the Perron-Frobenius theorem. The key idea is to replace the bilinear functional $x^T A y$ (or quadratic form $x^T A x$) with the multilinear functional (or homogeneous polynomial) associated with a tensor (or symmetric tensor). The constrained critical values/points then yield a notion of singular values/vectors (or eigenvalues/vectors) for order-k tensors. An important distinction between order-2 and order-k cases is the choice of norm for the constraints. For order-2 tensors, the $l^2$-norm is typically used, but for order-k tensors, the $l^k$-norm is needed to preserve scale invariance. The paper defines eigenpairs and singular pairs of tensors with respect to any $l^p$-norm (p > 1) and shows that for p = 2, the defining equations for singular values/vectors correspond to the equations obtained in the best rank-1 approximation of tensors. For symmetric tensors, the equations for eigenvalues/vectors for p = 2 and p = k define Z-eigenvalues/vectors and H-eigenvalues/vectors, respectively. The paper also discusses the applications of these concepts, including the use of $l^k$-eigenvalues for characterizing the positive definiteness of homogeneous polynomial forms. It presents a theorem showing that a positive irreducible tensor has a positive real $l^k$-eigenvalue with a corresponding $l^k$-eigenvector with all non-negative entries. The paper also proposes a multilinear generalization of the Perron-Frobenius theorem and discusses its application in ranking linked objects and studying hypergraphs.This paper proposes a theory of eigenvalues, eigenvectors, singular values, and singular vectors for tensors using a constrained variational approach similar to the Rayleigh quotient for symmetric matrices. The approach generalizes the spectral theory of matrices to higher-order tensors. For example, it discusses a multilinear generalization of the Perron-Frobenius theorem. The key idea is to replace the bilinear functional $x^T A y$ (or quadratic form $x^T A x$) with the multilinear functional (or homogeneous polynomial) associated with a tensor (or symmetric tensor). The constrained critical values/points then yield a notion of singular values/vectors (or eigenvalues/vectors) for order-k tensors. An important distinction between order-2 and order-k cases is the choice of norm for the constraints. For order-2 tensors, the $l^2$-norm is typically used, but for order-k tensors, the $l^k$-norm is needed to preserve scale invariance. The paper defines eigenpairs and singular pairs of tensors with respect to any $l^p$-norm (p > 1) and shows that for p = 2, the defining equations for singular values/vectors correspond to the equations obtained in the best rank-1 approximation of tensors. For symmetric tensors, the equations for eigenvalues/vectors for p = 2 and p = k define Z-eigenvalues/vectors and H-eigenvalues/vectors, respectively. The paper also discusses the applications of these concepts, including the use of $l^k$-eigenvalues for characterizing the positive definiteness of homogeneous polynomial forms. It presents a theorem showing that a positive irreducible tensor has a positive real $l^k$-eigenvalue with a corresponding $l^k$-eigenvector with all non-negative entries. The paper also proposes a multilinear generalization of the Perron-Frobenius theorem and discusses its application in ranking linked objects and studying hypergraphs.