Topological states of quantum matter such as topological insulators and superconductors have been an active field of research in physics for many years. The recent experimental progress on their realization with ultracold atomic gases raises natural questions about the notion of topological quantum matter far from thermal equilibrium. In this talk, we present our theoretical findings in this context.

We first discuss the non-equilibrium Hall response of a system initialized in a topologically trivial state before its Hamiltonian is ramped into a Chern insulator phase, comparing the coherent dynamics with effects of dephasing. In the second part of the talk, we discuss non-equilibrium properties of driven systems. In particular, we show how a perfect spin momentum locking, so far only known from edge states of two-dimensional topological insulators can emerge in the stroboscopic dynamics of one-dimensional Floquet lattice systems. Finally, we discuss dynamically defined topological quantum numbers that have no non-equilibrium counterpart, and report on their very recent experimental observation.