Recent observations indicate that in a large fraction of binary neutron star (BNS) mergers a long-lived neutron star (NS) may be formed rather than a black hole. Unambiguous electromagnetic (EM) signatures of such a scenario would strongly impact our knowledge on how short gamma-ray bursts (SGRBs) and their afterglow radiation are generated. Furthermore, such EM signals would have profound implications for multimessenger astronomy with joint EM and gravitational-wave (GW) observations of BNS mergers, which will soon become reality with the next science runs of the advanced LIGO/Virgo network of ground-based GW detectors. I will discuss BNS merger scenarios involving a long-lived NS (including the so-called ``time-reversal" scenario) and present a detailed model to follow the post-merger evolution of the system and to predict its EM emission. Results will be discussed in the context of SGRBs, their X-ray afterglows, and multimessenger astronomy.