Propagator models describe transient price impact by allowing execution prices to depend on the entire history of past trades through a Volterra kernel. While such models capture empirically observed features of market microstructure, they lead to infinite-dimensional and non-Markovian optimisation problems that fall outside the scope of classical arbitrage and duality theory. In this talk, I develop a comprehensive mathematical finance framework for propagator models, including a fundamental theorem of asset pricing, a superreplication theorem with liquidity-adjusted risk measures, and a convex-duality approach to utility maximisation. Despite non-linear preferences and path-dependent price impact, optimal strategies are obtained by solving a suitably constructed dual problem over a family of fictitious frictionless financial markets.