The studies of scattering amplitudes during recent years has uncovered many new features of
various theories that are invisible from the action point of view. One of the most interesting
features is the relation between gauge theories and gravity theories, known as double copy. The
most famous example is the relation between pure Yang-Mills theory and axion-dilaton gravity,
however over the years many more examples of this relation have been found. It is interesting
to understand what is special about these theories that can be related by double copy. Most
of the known examples of such theories have massless force carriers so a natural question is
can double copy be extended to massive theories? The work presented in this thesis suggests
that such massive theories must be strongly constrained since a naive attempt to double copy a
generic massive gauge theory leads to unphysical scattering amplitudes. In 4d the only known
examples are Kaluza-Klein theories of 5d massless gauge theories. However in 3d the situation is
different. The work of this thesis suggests that topologically massive gauge and gravity theories
are related by double copy. We observed this relation between scattering amplitudes (for 3,4
and 5 point tree level amplitudes) as well as between classical solutions. Also, we found a 3d
equivalent of Weyl double copy relating the Cotton tensor to a product of two gauge theory
field strength tensors.