Dynamical properties of a many-body system are determined by its properties as a quantum bath: the systems that thermalize act as an efficient bath, while integrable and many-body localized (MBL) systems fail to do so. I will describe a new approach to quantum many-body dynamics, inspired by the notion of the Feynman-Vernon influence functional (IF). I will consider interacting spin systems, and formulate an equation satisfied by their influence functionals. While difficult in general, this equation can be solved exactly for a class of many-body systems – perfect dephasers – which act as Markovian baths on their subsystems. More generally, I will show that, viewed as a fictitious wave function in the temporal domain, influence functional can be described by tensor-network methods. This approach is based on the behavior of temporal entanglement of the IF, which remains relatively low in very different physical regimes, including fast thermalization, integrability, and many-body localization. IF approach offers a new lens on many-body non-equilibrium phenomena, both in ergodic and non-ergodic regimes, connecting the theory of open quantum systems to quantum statistical physics.