Summary: We give upper bounds for the number of vertices of the polyhedron \(P (A,b) = \{ x \in \mathbb R^d : Ax \leqslant b\}\) when the \(m\times d\) constraint matrix \(A\) is subjected to certain restriction. For instance, if \(A\) is a 0/1-matrix, then there can be at most \(d\)! vertices and this bound is tight, or if the entries of \(A\) are non-negative integers so that each row sums to at most \(C\), then there can be at most \(C^d\) vertices. These bounds are consequences of a more general theorem that the number of vertices of \(P(A,b)\) is at most \(d\)!\(\bullet W/D\), where \(W\) is the volume of the convex hull of the zero vector and the row vectors of \(A\), and \(D\) is the smallest absolute value of any non-zero \(d\times d\) subdeterminant of \(A\).