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Recent work showed that $\kappa$-deformations can describe the quantumdeformation of several relativistic models that have been proposed in thecontext of quantum gravity phenomenology. Starting from the Poincar\'e algebraof special-relativistic symmetries, one can toggle the curvature parameter$\Lambda$, the Planck scale quantum deformation parameter $\kappa$ and thespeed of light parameter $c$ to move to the well-studied $\kappa$-Poincar\'ealgebra, the (quantum) (A)dS algebra, the (quantum) Galilei and Carrollalgebras and their curved versions. In this review, we survey the propertiesand relations of these algebras of relativistic symmetries and their associatednoncommutative spacetimes, emphasizing the nontrivial effects of interplaybetween curvature, quantum deformation and speed of light parameters.

Interplay between spacetime curvature, speed of light and quantum deformations of relativistic symmetries