I’ll discuss the use of small cancellation methods in producing groups with desired properties. In particular, I’ll demonstrate how to construct groups that satisfy certain laws with probability 1 but still contain free non-abelian subgroups (answering a question of Amir, Blachar, Gerasimova, and Kozma); groups whose semigroup Zariski topology is strictly coarser than their group Zariski topology (answering a question of Elliott, Jonusas, Mesyan, Mitchell, Morayne, and Peresse); and lacunary hyperbolic groups with very small Frattini quotients (in contrast to previously known results by Gelander and Glasner, and by Hull, for other hyperbolic-like groups).