  This text is an introduction to the author's cohomological approach, based on
Hodge theory, to (effective) unique ergodicity and weak mixing of translation
flows. Compared to earlier expositions, it emphasizes the analogy between the
two problems by introducing a point of view on weak mixing based on the
appropriate twisted cohomology. In particular, a new cohomological proof, based
on Hodge theory for the twisted cohomology, of Veech's classical criterion for
weak mixing is presented here for the first time. The exposition also aims to
give a general introduction to the ergodic theory of translation flows and
related systems and includes references to background material and related
developments in the theory, as well as several exercises, open problems and
conjectures.
