In this paper we develop the first Compressive Sensing (CS) adaptive radar detector. We propose three novel architectures and demonstrate how a classical Constant False Alarm Rate (CFAR) detector can be combined with l 1 -norm minimization. Using asymptotic arguments and the Complex Approximate Message Passing (CAMP) algorithm we characterize the statistics of the l 1 -norm reconstruction error and derive closed form expressions for both the detection and false alarm probabilities. We support our theoretical findings with a range of experiments that show that our theoretical conclusions hold even in non-asymptotic setting. We also report on the results from a radar measurement campaign, where we designed ad hoc transmitted waveforms to obtain a set of CS frequency measurements. We compare the performance of our new detection schemes using Receiver Operating Characteristic (ROC) curves.