  We prove that the topological complexity of every symplectically atoroidal
manifold is equal to twice its dimension. This is the analogue for topological
complexity of a result of Rudyak and Oprea, who showed that the
Lusternik--Schnirelmann category of a symplectically aspherical manifold equals
its dimension. Symplectically hyperbolic manifolds are symplectically
atoroidal, as are symplectically aspherical manifolds whose fundamental group
does not contain free abelian subgroups of rank two. Thus we obtain many new
calculations of topological complexity, including iterated surface bundles and
symplectically aspherical manifolds with hyperbolic fundamental groups. Our
result also applies in the greater generality of cohomologically symplectic
manifolds.
