**Dynamics and Topological States in Strongly-Coupled Light-Matter Systems**

Recently,  communities of condensed-matter physics and atonic physics physics working with a strong element of light-matter coupling have reached new limits towards realizing novel dynamical and topological states of matter. In this presentation, we review (our) theoretical efforts to achieve these goals addressing new time-dependent phenomena related to Floquet theory, generalized NMR protocols and related to these current experimental realizations.  At the same time, these developments address new questions related to measurements of topological invariants in the time or frequency domain. We present our efforts to address topology in circuit Quantum Electrodynamics, Josephson junction architectures and ultra-cold atoms in the strong-interaction limit. Topology can be (already) addressed at a classical level with simple LC elements. We also report a quantum dynamo effect on a sphere, when rotating a spin-1/2 (subject to a radial magnetic field) strongly entangled with an environment and measuring the Berry phase of this spin-1/2.