Acquisition times in magnetic resonance imaging (MRI) are typically in the order of minutes. For an image of 256 x 256 pixels, the standard Fourier reconstruction technique used in most commercial imaging systems requires 256 separate free induction decay (FID) signals. While the FID signal itself is of relatively short duration, the successive FID signals are separated by long delays, of the order of seconds, to permit substantial relaxation of the signal before the next excitation. The resultant long acquisition times give rise to motion artefacts, preclude dynamic imaging and keep the patient throughput low. In this paper, we investigate a fast imaging scheme which uses spiral trajectories in the spatial frequency domain. The entire domain can be sampled in a short time, requiring as few as one FID acquisition. The scheme requires time varying gradients having the form of a ramped sinusoid. Several reconstruction methods are considered for forming the image from the spatial frequency domain data. The possibility of using multiple spirals to deal with the rapid decay of the FID signal is also examined in detail.