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Eigenvectors of tensors, an extension of eigenvectors of square matrices, were intro-duced by L.-H. Lim and L. Qi independently in 2005 and have been studied in numerical multi-linear algebra. Recently, the concept of eigenvectors of a tensor drew attention to the algebraic geometry community because algebraic geometry is proven to provide useful techniques for the tensor eigenproblem. The purpose of this talk is to discuss algebra-geometric aspects of tensor eigenvectors, which include an algebro-geometric characterization of tensor eigenvectors and the discriminant of a set of tensor eigenvec-tors. This talk is base on joint work with A. Seigal and B. Sturmfels.