  We develop operator renewal theory for flows and apply this to obtain results
on mixing and rates of mixing for a large class of finite and infinite measure
semiflows. Examples of systems covered by our results include suspensions over
parabolic rational maps of the complex plane, and nonuniformly expanding
semiflows with indifferent periodic orbits. In the finite measure case, the
emphasis is on obtaining sharp rates of decorrelations, extending results of
Gou\"ezel and Sarig from the discrete time setting to continuous time. In the
infinite measure case, the primary question is to prove results on mixing
itself, extending our results in the discrete time setting. In some cases, we
obtain also higher order asymptotics and rates of mixing.
