Flat bands allow to increase the critical temperature of the superconducting transition thanks to their high density of states. However a characterization of the flat bands that support a finite supercurrent is open. In some cases a nonzero Chern number, a topological invariant of the band structure, ensures a finite superfluid mass density. This tantalizing relation between topology and superfluidity is novel and unexplored. My aim is to characterize superfluidity in lattice systems with flat bands that have different symmetries, lattice structures, dimensionality, interparticle interactions and possess different topological invariants, in order to provide a general picture of which ones are potentially useful as a superconductor with high critical temperature. Whereas mean-field (BCS) theory can provide an essential qualitative understanding and a transparent link to topological properties, I plan to use more reliable methods such as Density Matrix Renormalization Group (DMRG) in 1D and Dynamical Mean Field Theory (DMFT) in 2D and 3D. The ideal platform to test the theoretical predictions are ultracold gases, but I expect to provide useful results also for multiband superconductors, topological media, carbon-based superconductors, Quantum Hall systems and high-Tc superconductors.