In this monograph we lay the foundation for a theory of coarse groups and
coarse actions. Coarse groups are group objects in the category of coarse
spaces, and can be thought of as sets with operations that satisfy the group
axioms "up to uniformly bounded error". In the first part of this work, we
develop the theory of coarse homomorphisms, quotients, and subgroups, and prove
that coarse versions of the Isomorphism Theorems hold true. We also initiate
the study of coarse actions and show how they relate to the fundamental
observation of Geometric Group Theory.
In the second part we explore a selection of specialized topics, such as the
study of coarse group structures on set-groups, groups of coarse automorphisms
and spaces of controlled maps. Here the main aim is to show how the theory of
coarse groups connects with classical subjects. These include: number theory;
the study of bi\=/invariant metrics on groups; quasimorphisms and stable
commutator length; ${\rm Out}(F_n)$; topological group actions.