  Toric varieties are a special class of rational varieties defined by
equations of the form {\it monomial = monomial}. For a good brief survey of the
history and role of toric varieties see [10]. Any toric variety $X$ contains a
cover by affine open sets described in terms of arrangements (called fans) of
convex bodies in $\Bbb R^r$. The coordinate rings of each of these affine open
sets is a graded ring generated over the ground field by monomials. As a
consequence, toric varieties provide a good context in which cohomology can be
calculated. The purpose of this article is to describe the second \'etale
cohomology group with coefficients in the sheaf of units of any toric variety
$X$. This is the so-called cohomological Brauer group of $X$.
