Complex quantum many-body systems are ubiquitous in nature, yet their behaviour often remains very challenging to predict with analytical or numerical calculations - especially when it comes to dynamics. However, using ultracold atoms in optical lattices it is possible to create precisely tunable, yet very accessible complex systems, which can be probed with a variety of observables. Using this experimental set-up, we demonstrate how a periodically modulated system can be described by an effective Floquet-Hamiltonian on longer time-scales - even when driving the system far from equilibrium. This allows for implementing Haldane's model for a topological insulator by applying an oscillating force to a honeycomb lattice, and mapping out its topological transitions. We also extend this approach to spin-dependent and interacting systems. Furthermore we present how the distribution of anti-ferromagnetic correlations in the Hubbard model depend on the geometry of the lattice. We also study how correlations re-arrange as a function of time by deforming the lattice geometry on time-scales ranging from the sudden to the adiabatic regime.