The first multimessenger observation of a binary neutron star (BNS) merger in 2017 confirmed that such a system can launch a powerful jet and produce a short gamma-ray burst (SGRB). Theoretical modelling of the involved physical processes via numerical simulations is however extremely challenging. Despite the great advancements in general relativistic magnetohydrodynamic simulations of BNS mergers, in particular over the last decade, the jet formation mechanism and the nature of the central engine (massive neutron star vs. accreting black hole) remain unclear. Moreover, the propagation of the incipient jet across the environment surrounding the merger site up to the scales relevant for the electromagnetic emission is typically modelled separately from the merger process, without a consistent end-to-end description. In this talk, I will review recent developments in simulations of BNS mergers and SGRB jet production and propagation, along with future prospects and upcoming challenges.