MoEDAL is a pioneering LHC experiment designed to search for anomalously ionizing messengers of new physics, such as the magnetic monopole. After a test run at 8 TeV centre-of-mass energy (Ecm), it started official data taking at the LHC at an Ecm of 13 TeV, in 2015. Its groundbreaking physics program defines a number of scenarios that yield potentially revolutionary insights into such foundational questions as: are there extra dimensions or new symmetries; what is the mechanism for the generation of mass; does magnetic charge exist; do topological particles exist; and what is the nature of dark matter? After a brief introduction, MoEDAL's progress to date will be reported, including its past, current and expected future physics output. Additionally, an upgrade to the MoEDAL detector... consisting of two new subdetectors: MAPP (MoEDAL Apparatus for Penetrating Particles) now being prototyped at IP8; and MALL (MoEDAL Apparatus for very long-lived particles), will be presented. Finally, a possible astroparticle extension to MoEDAL, called Cosmic-MoEDAL, will be briefly described. This high altitude detector will allow the search for magnetic monopoles to be continued from the TeV scale to the GUT scale.This article is part of a discussion meeting issue ‘Topological avatars of new physics’.