The Line Emission Mapper (LEM) is a Probe mission concept developed in response to NASA's Astrophysics Probe Explorer (APEX) Announcement of Opportunity. LEM has a single science instrument composed of a large-area, wide-field X-ray optic and a microcalorimeter X-ray imaging spectrometer in the focal plane. LEM is optimized to observe low-surface-brightness diffuse X-ray emission over a 30 ′ equivalent diameter field of view with 1.3 and 2.5 eV spectral resolution in the 0.2 − 2.0 keV band. Our primary scientific objective is to map the thermal, kinetic, and elemental properties of the diffuse gas in the extended X-ray halos of galaxies, the outskirts of galaxy clusters, the filamentary structures between these clusters, the Milky Way star-formation regions, the Galactic halo, and supernova remnants in the Milky Way and Local Group. The combination of a wide-field optic with 18 ′′ angular resolution end-to-end and a microcalorimeter array with 1.3 eV spectral resolution in a 5 ′ × 5 ′ inner array (2.5 eV outside of that) offers unprecedented sensitivity to extended low-surface-brightness X-ray emission. This allows us to study feedback processes, gas dynamics, and metal enrichment over seven orders of magnitude in spatial scales, from parsecs to tens of megaparsecs. LEM will spend approximately 11% of its five-year prime science mission performing an All-Sky Survey, the first all-sky X-ray survey at high spectral resolution. The remainder of the five-year science mission will be divided between directed science (30%) and competed General Observer science (70%). LEM and the NewAthena/XIFU are highly complementary,