There is no pre-seminar.

We introduce the Tesler polytope ${\rm{TP}}_n({\bf a})$, whose integer points are the Tesler matrices of size $n$ with hook sums $a_1,a_2,\ldots,a_n \in \mathbb{Z}_{\geq 0}$. We show that ${\rm{TP}}_n({\bf a})$ is a flow polytope and therefore the number of Tesler matrices is counted by the type $A_n$ Kostant partition function evaluated at $(a_1,a_2,\ldots,a_n,{-\sum_{i=1}^n} a_i)$. We describe the faces of this polytope in terms of ``Tesler tableaux'' and characterize when the polytope is simple. We prove that the $h$-vector of ${\rm{TP}}_n({\bf a})$ when all $a_i>0$ is given by the Mahonian numbers and calculate the volume of ${\rm{TP}}_n(1,1,\ldots,1)$ to be a product of consecutive Catalan numbers multiplied by the number of standard Young tableaux of staircase shape.