We consider the generalization of the Birkhoff’s theorem in higher dimensions. An *n*×*n*×*n* stochastic tensor is a nonnegative array (hypermatrix) in which every sum over one index is 1. A permutation tensor can be identified with a Latin square (vice versa). We study the polytope of all these tensors, the convex set of all tensors with some positive diagonals, and the polytope generated by the permutation tensors. We present lower and upper bounds for the number of vertices of the polytopes, and discuss further questions on the topic.

Determinant and permanent are basic and important functions of *n* × *n* matrices. We attempt to define these for tensors. More generally, we will consider defining the generalized matrix functions for tensors.