A novel fast and quiet method of magnetic resonance imaging (MRI) is introduced which creates new opportunities for imaging in medicine and materials science. The method is called SWIFT, sweep imaging with Fourier transformation. In SWIFT, time-domain signals are acquired in a time-shared manner during a swept radiofrequency excitation of the nuclear spins. With negligible time between excitation and signal acquisition, new possibilities exist for imaging objects consisting of spins with extremely fast transverse relaxation rates, such as macromolecules, semi-solids, and quadrupolar nuclei. The field gradient used for spatial-encoding is not pulsed on and off, but rather is stepped in orientation in an incremental manner, which results in low acoustic noise. This unique acquisition method is expected to be relatively insensitive to sample motion, which is important for imaging live objects. Additionally, the frequency-swept excitation distributes the signal energy in time and thus dynamic range requirements for proper signal digitization are reduced compared with conventional MRI. For demonstration, images of a plastic object and cortical bone are shown.