In this paper, we study T-eigenvalues of third-order tensors. Definitions of the T-eigenvalues and Hermitian tensors are proposed. We present a commutative tensor family. We prove some T-eigenvalue inequalities for Hermitian tensors, including extensions of Weyl's theorem and Cauchy's interlacing theorem from the matrix case to the tensor case. Finally, we introduce the stability of the T-eigenvalues and study the Lyapunov equation for tensors.