The unprecedented collision energy of the LHC has opened up a new discovery frontier, however, without any signs of new physics in sight. The first LHC dedicated search experiment, MoEDAL, started data taking for Run 2. MoEDAL is designed to search highly ionising particle avatars of new physics using $pp$ and heavy-ion collisions at the LHC. The planned upgrade for MoEDAL at Run 3 of the LHC $-$ the MAPP detector (MoEDAL Apparatus for Penetrating Particles) $-$ will extend MoEDAL's physics reach to include feebly interacting, long-lived messengers of physics beyond the Standard Model. This will allow the experiment to explore a number of models of new physics, including dark sector models, in a complementary way to that of conventional LHC collider experiment detectors. Furthermore, a possible astroparticle extension to MoEDAL, called Cosmic-MoEDAL, will allow the search for magnetic monopoles to be continued from the TeV scale to the GUT scale. This paper focuses on recent results and plans for the LHC Run 3.