  We describe structure of quasihomomorphisms from arbitrary groups to discrete
groups. We show that all quasihomomorphisms are 'constructible', i.e., are
obtained via certain natural operations from homomorphisms to some groups and
quasihomomorphisms to abelian groups. We illustrate this theorem by describing
quasihomomorphisms to certain classes of groups. For instance, every unbounded
quasihomomorphism to a torsion-free hyperbolic group H is either a homomorphism
to a subgroup of H or is a quasihomomorphism to an infinite cyclic subgroup of
H.
