  We study torsion in the integral cohomology of a certain family of
$2n$-dimensional orbifolds $X$ with actions of the $n$-dimensional compact
torus. Compact simplicial toric varieties are in our family. For a prime number
$p$, we find a necessary condition for the integral cohomology of $X$ to have
no $p$-torsion. Then we prove that the necessary condition is sufficient in
some cases. We also give an example of $X$ which shows that the necessary
condition is not sufficient in general.
