We introduce a new approach to radar imaging based on the concept of compressive sensing (CS). In CS, a low-dimensional, nonadaptive, linear projection is used to acquire an efficient representation of a compressible signal directly using just a few measurements. The signal is then reconstructed by solving an inverse problem either through a linear program or a greedy pursuit. We demonstrate that CS has the potential to make two significant improvements to radar systems: (i) eliminating the need for the pulse compression matched filter at the receiver, and (ii) reducing the required receiver analog-to-digital conversion bandwidth so that it need operate only at the radar reflectivity's potentially low "information rate" rather than at its potentially high Nyquist rate. These ideas could enable the design of new, simplified radar systems, shifting the emphasis from expensive receiver hardware to smart signal recovery algorithms.