This talk is an introduction to coarse homomorphisms. Morally, a coarse homomorphism (or quasimorphism) between two groups is a function that sends the product of any two elements within uniformly bounded distance from the product of their images. The study of quasimorphisms from a group G to the real numbers is a relatively classical topic, that has connection with (stable) commutator length and bounded cohomology on groups. The aim of this talk is to show that by considering coarse homomorphisms with more general target groups naturally lead us to the notions of coarse groups and subgroups. Throughout the talk, special emphasis will be given to the coarse homomorphisms with target on a Banach space.