The $\kappa$-deformation which turns the Poincar{\'e} algebra into a quantum group has been widely studied over the past 30 years. The main reason is that the $\kappa$-Poincar{\'e} algebra describes the symmetries of a quantum spacetime which arises as a limit of quantum gravity. Although the classical properties of field theory built on this quantum spacetime have been widely studied, it was only recently that significant progress was made in the understanding of their quantum properties. After a general introduction, I will present the computation of the one-loop 2-point and 4-point functions for various models of $\kappa$-Poincar{\'e} invariant scalar field theory with quartic interactions whose low-energy limit ($\kappa\to\infty$) coincides with the ordinary $\phi^4$-model on Minkowski spacetime. The computation is performed using a star product obtained from mere adaptation of the Weyl quantisation scheme. For one model, the one-loop 4-point function is found to be finite. As a major conceptual result, I will show that requiring the $\kappa$-Poincar{\'e} invariance forces the integral involved in the action functional to be a twisted trace, thus defining a KMS weight for the C*-algebra modelling the quantum spacetime. Finally, phenomenological implications of these results, together with the possibility to constrain these models by exploiting cosmological data -- such as $\gamma$-ray burst --, will be discussed.
