  We prove a complexity dichotomy theorem for the eight-vertex model. For every
setting of the parameters of the model, we prove that computing the partition
function is either solvable in polynomial time or \#P-hard. The dichotomy
criterion is explicit. For tractability, we find some new classes of problems
computable in polynomial time. For \#P-hardness, we employ M\"{o}bius
transformations to prove the success of interpolations.
