I will discuss our recently developed approaches to understanding the electronic structure of solids under pump-probe conditions, in particular, how in the Floquet-limit the concept of a band structure can be extended to time-dependent systems and under which conditions it remains an observable of  photo-electron spectroscopy.  I will show how the calculated pump- 
probe photo-electron spectrum of WSe2 shows non-equilibrium feat-ures that are the fingerprint of a steady-state non-equilibrium phase of the material. Such non-equilibrium systems can exhibit a variety of interesting properties, some of which can be funda-mentally different from their equi-librium properties. In this context, I will discuss the possibility how laser pulses can be used to switch between topological phases in a prototypical 3D Dirac material, Na3Bi, and I will demonstrate how lasers can enhance topological protection against lattice perturb-ations. Finally, I will discuss how the excitation of a coherent phonon mode modifies the photo-electron spectrum by electron-phonon coupling.

Figure: Intersection cones in the Floquet band structure of periodically driven Na3Bi forming Floquet-Weyl points