  In this contribution we first summarize how contour integration methods can
be used to derive closed formulae for functional determinants of ordinary
differential operators. We then generalize our considerations to partial
differential operators. Examples are used to show that also in higher
dimensions closed answers can be obtained as long as the eigenvalues of the
differential operators are determined by transcendental equations. Examples
considered comprise of the finite temperature Casimir effect on a ball and the
functional determinant of the Laplacian on a two-dimensional torus.
