It is possible that relativistic symmetries become deformed in the semiclassical regime of quantum gravity. Mathematically, such deformations lead to the noncommutativity of spacetime geometry and non-vanishing curvature of momentum space. The best studied example is given by the $\kappa$-Poincaré Hopf algebra, associated with $\kappa$-Minkowski space. On the other hand, the curved momentum space is a well-known feature of particles coupled to three-dimensional gravity. The purpose of this thesis was to explore some properties and mutual relations of the above two models. In particular, I study extensively the spectral dimension of $\kappa$-Minkowski space. I also present an alternative limit of the Chern-Simons theory describing three-dimensional gravity with particles. Then I discuss the spaces of momenta corresponding to conical defects in higher dimensional spacetimes. Finally, I consider the Fock space construction for the quantum theory of particles in three-dimensional gravity.