What are some applications of computing the permanent of a matrix?
Valiant proved that the permanent is $\# P$-complete, which means that an efficient algorithm for computing the permanent can be used to solve any problem in $\# P$, such as counting the number of satisfying assignment to a CNF, the number of Hamiltonian circuits, the number of $k$-colorings and so on. In particular, it could be used to solve NP-complete problems.