Weinstein domains and their symplectic invariants have been extensively studied over the last 30 years. Little is known about non-Weinstein Liouville domains, whose first instance is due to McDuff. I will describe two key examples of such domains in dimension four, and then explain how they fit into a general construction based on Anosov flows on three-manifolds. The symplectic invariants of these “Anosov Liouville domains” constitute new invariants of Anosov flows. The algebraic structure of their wrapped Fukaya categories is in stark contrast with the Weinstein case.   This is mostly based on joint work arXiv:2211.07453 with Kai Cieliebak, Oleg Lazarev and Agustin Moreno