This colloquium offers a gentle introduction to the physics of topological materials using vivid mechanical demonstrations. The distinctive property of topological materials is the existence of states and excitations that are robust (or protected) against structural deformations, changes in material parameters or imperfections. We concentrate on two examples of topologically protected states: the folding motions of origami-like structures and sound propagation in active fluids composed of self-propelled particles. In both cases we trace the mathematical origin of physical robustness to elegant notions of topology.