  We review the Schwinger-Keldysh, or in-in, formalism for studying quantum
dynamics of systems out-of-equilibrium. The main motivation is to rephrase well
known facts in the subject in a mathematically elegant setting, by exhibiting a
set of BRST symmetries inherent in the construction. We show how these
fundamental symmetries can be made manifest by working in a superspace
formalism. We argue that this rephrasing is extremely efficacious in
understanding low energy dynamics following the usual renormalization group
approach, for the BRST symmetries are robust under integrating out degrees of
freedom. In addition we discuss potential generalizations of the formalism that
allow us to compute out-of-time-order correlation functions that have been the
focus of recent attention in the context of chaos and scrambling. We also
outline a set of problems ranging from stochastic dynamics, hydrodynamics,
dynamics of entanglement in QFTs, and the physics of black holes and cosmology,
where we believe this framework could play a crucial role in demystifying
various confusions. Our companion paper arXiv:1610.01941 describes in greater
detail the mathematical framework embodying the topological symmetries we
uncover here.
