/ AbstractThe Line Emission Mapper (LEM) is an X-ray Probe with with spectral resolution ~2 eV FWHM from 0.2 to 2.5 keV and effective area >2,500 cm$^2$ at 1 keV, covering a 33 arcmin diameter Field of View with 15 arcsec angular resolution, capable of performing efficient scanning observations of very large sky areas and enabling the first high spectral resolution survey of the full sky. The LEM-All-Sky Survey (LASS) is expected to follow the success of previous all sky surveys such as ROSAT and eROSITA, adding a third dimension provided by the high resolution microcalorimeter spectrometer, with each 15 arcsec pixel of the survey including a full 1-2 eV resolution energy spectrum that can be integrated over any area of the sky to provide statistical accuracy. Like its predecessors, LASS will provide both a long-lasting legacy and open the door to the unknown, enabling new discoveries and delivering the baseline for unique GO studies. No other current or planned mission has the combination of microcalorimeter energy resolution and large grasp to cover the whole sky while maintaining good angular resolution and imaging capabilities. LASS will be able to probe the physical conditions of the hot phases of the Milky Way at multiple scales, from emission in the Solar system due to Solar Wind Charge eXchange, to the interstellar and circumgalactic media, including the North Polar Spur and the Fermi/eROSITA bubbles. It will measure velocities of gas in the inner part of the Galaxy and extract the emissivity of the Local Hot Bubble. By maintaining the original angular resolution, LASS will also be able to study classes of point sources through stacking. For classes with ~$10^4$ objects, it will provide the equivalent of 1 Ms of high spectral resolution data. We describe the technical specifications of LASS and highlight the main scientific objectives that will be addressed. (Abridged)