Critical mechanical structures are structures at the verge of mechanical instability. They are characterized by their floppy modes, which are deformations costing little energy. On the one hand, numerous interesting phenomena in soft matter are governed by the physics of critical mechanical structures, because they capture the critical state between solid and liquid. On the other hand, the design of mechanical metamaterials (i.e., engineered materials that gain their unusual mechanical properties, such as negative Poisson's ratio, from their structures) often rely on floppy modes to realize novel properties. In this talk, we focus on floppy modes in critical mechanical structures that are governed by topological invariants, and are thus called "topological edge modes.” We discuss how intrinsic soft strains of periodic critical mechanical structures can be used to induce topological transitions, leading to dramatic changes in mechanical properties.