The LMT is a 50 m diameter telescope for operation to a wavelength of 1 mm from a high mountain site in central Mexico at a latitude of 19 degrees. The telescope is designed to address fundamental questions about the origin and formation of galaxies, clusters of galaxies as well as stars and planets. It is a joint project of the Mexican Instituto Nacional de Astrofisica, Optica y Electronica (INAOE) in Tonantzintla, Puebla and the University of Massachusetts at Amherst. The telescope is in the Critical Design phase and completion of the basic assembly at the site is scheduled for the end of 2000. We describe historical and organizational aspects of the Project and present the major technical specifications and plans for realization.