  In this paper we review some aspects of relativistic particles' mechanics in
the case of a non-trivial geometry of momentum space. We start with showing how
the curved momentum space arises in the theory of gravity in 2+1 dimensions
coupled to particles, when (topological) degrees of freedom of gravity are
solved for. We argue that there might exist a similar topological phase of
quantum gravity in 3+1 dimensions. Then we characterize the main properties of
the theory of interacting particles with curved momentum space and the
symmetries of the action. We discuss the spacetime picture and the emergence of
the principle of relative locality, according to which locality of events is
not absolute but becomes observer dependent, in the controllable, relativistic
way. We conclude with the detailed review of the most studied kappa-Poincare
framework, which corresponds to the de Sitter momentum space.
