Conventional radar transmits electromagnetic waves towards the targets of interest. In between the outgoing pulses, the radar measures the signal reflected from the targets to determine their presence, range, velocity and other characteristics. Radar systems face multiple challenges, generating many trade-offs such as bandwidth versus range resolution and dwell time versus Doppler resolution. In MIMO radar, high resolution requires a large aperture and high number of antennas, increasing hardware and processing requirements. Recently, novel approaches in sampling theory and radar signal processing have been proposed to allow target detection and parameter recovery from samples obtained below the Nyquist rate. These techniques exploit the sparsity of the target scene in order to reduce the required number of samples, pulses and antennas, breaking the link between bandwidth, dwell time and number of antennas on the one hand and range, Doppler and azimuth resolution, respectively, on the other. This review introduces this so-called sub-Nyquist radar paradigm and describes the corresponding sampling and recovery algorithms, that leverage compressed sensing techniques to perform time and spatial compression. We focus on non radar imaging applications and survey many recent compressed radar systems. Our goal is to review the main impacts of compressed radar on parameter resolution as well as digital and analog complexity. The survey includes fast and slow time compression schemes as well as spatial compression approaches. We show that beyond substantial rate reduction, compression may also enable communication and radar spectrum sharing. Throughout the paper, we consider both theoretical and practical aspects of compressed radar, and present hardware prototype implementations, demonstrating real-time target parameter recovery from low rate samples in pulse-Doppler and MIMO radars.