the entire sky visible from The Netherlands (about 60% of the entire sky) in only one night. Transient radio phenomena, only hinted at by previous narrow-field surveys, will be discovered, rapidly localised with unprecedented accuracy, and automatically compared to data from other facilities (e.g. gamma-ray, optical, X-ray observatories). Such transient phenomena may be associated with exploding stars, black holes, flares on Sun-like stars, radio bursts from exoplanets or even SETI signals. In addition, this key science project will make a deep survey for radio pulsars at low radio frequencies, and will attempt to detect giant radio bursts from rotating neutron stars in distant galaxies. Ultra high-energy cosmic rays LOFAR offers a unique possibility in particle physics for studying the origin of high-energy and ultra-high-energy cosmic rays (HECRs and UHECRs) at energies between eV. Both the sites and processes for accelerating particles are unknown. Possible candidate sources of these HECRs are shocks in radio lobes of powerful radio galaxies, intergalactic shocks created during the epoch of galaxy formation, so-called Hyper-novae, gamma-ray bursts, or decay products of super-massive particles from topological defects, left over from phase transitions in the early Universe. The primary observable is the intense radio pulse that is produced when a primary CR hits the atmosphere and produces an extensive air shower (EAS). An EAS is aligned along the direction of motion of the primary particle, and a substantial part of its component consists of electron-positron pairs which emit radio emission in the terrestrial magnetosphere (e.g., geo-synchrotron emission). Cosmic magnetism LOFAR opens the window to the so far unexplored low-energy synchrotron radio waves, emitted by cosmic-ray electrons in weak magnetic fields. Very little is known about the origin and evolution of cosmic magnetic fields. The space around galaxies and between galaxies may all be magnetic, and LOFAR may be the first to detect weak radio emission from such regions. LOFAR will also measure the Faraday effect, which is the rotation of polarization plane of low-frequency radio waves, and gives another tool to detect weak magnetic fields. Solar physics and space weather The Sun is an intense radio source. The already strong thermal radiation of the K hot solar corona is superimposed by intense radio bursts that are associated with phenomena of the solar activity, like flares and coronal mass ejections (CMEs). Solar radio radiation in the LOFAR frequency range is emitted in the middle and upper corona. So LOFAR