  We compute the cohomology of the complement of toric arrangements associated
to root systems as representations of the corresponding Weyl groups.
Specifically, we develop an algorithm for computing the cohomology of the
complement of toric arrangements associated to general root systems and we
carry out this computation for the exceptional root systems $G_2$, $F_4$, $E_6$
and $E_7$. We also compute the total cohomology of the complement of the toric
arrangement associated to $A_n$ as a representation of the Weyl group and give
a formula for its Poincar\'e polynomial.
