The BEST Theorem (https://en.wikipedia.org/wiki/BEST_theorem) for the number of Eulerian circuits of an Eulerian directed graph $G$: $$ec(G) = t_w(G) \cdot \prod_{v\in V} (\mathrm{deg}(v)-1)!,$$ where $t_w(G)$ is the number of arborescences rooted at any fixed vertex $w\in G$. The number $t_w(G)$ can be computed as a determinant thanks to (a directed graph version of) the matrix-tree theorem, already mentioned in another answer.

This is a remarkable formula because, like many other formulas mentioned in answers to this question, it is right "on the border" of what is computationally tractable. For instance, as mentioned in the Wikipedia article above, the problem of counting Eulerian circuits in an undirected graph is by contrast #P-complete.

(Another very similar "on the border" result in enumeration in graph theory is the Kasteleyn method for computing perfect matchings of a planar graph, compared to the difficulty of computing perfect matchings of an arbitrary graph, which should be an answer if it is not already.)