The Atacama Large Millimeter Array (ALMA), a world-wide project (64x12m-dishes operating from 84 to 720 GHz, to be completed by 2011) will represent a jump of almost two orders of magnitude in sensitivity and angular resolution as compared to present millimeter/submillimeter interferometers, and will thus undoubtedly produce a major step in astrophysics. The main objectives will be the origins of galaxies, stars and planets. ALMA will be able to detect dust-enshrouded star-forming galaxies at redshifts z > 10, both in the emission of dust and spectral lines (CO and other species, including C+). It will also explore in detail the physical and chemical processes of star and planet formation hidden away in dusty molecular clouds and protoplanetary disks. In addition, ALMA will allow similar enormous gains in all other fields of mm and submm astronomy, including nearby galaxies, AGN, astrochemistry, circumstellar shells and the solar system.