Over the last two decades, dual-polarimetric weather radar has proven to be a valuable instrument providing critical precipitation information through remote sensing of the atmosphere. Modern weather radar systems operate with high sampling rates and long dwell times on targets. Often only limited target information is desired, leading to a pertinent question: could lesser samples have been acquired in the first place? Recently, a revolutionary sampling paradigm – compressed sensing (CS) – has emerged, which asserts that it is possible to recover signals from fewer samples or measurements than traditional methods require without degrading the accuracy of target information. CS methods have recently been applied to point target radars and imaging radars, resulting in hardware simplification advantages, enhanced resolution, and reduction in data processing overheads. But CS applications for volumetric radar targets such as precipitation remain relatively unexamined. This research investigates the potential applications of CS to radar remote sensing of precipitation. In general, weather echoes may not be sparse in space-time or frequency domain. Therefore, CS techniques developed for point targets, such as in aircraft surveillance radar, are not directly applicable to weather radars. However, precipitation samples are highly correlated both spatially and temporally. We, therefore, adopt latest advances in matrix completion algorithms to demonstrate the sparse sensing of weather echoes. Several extensions of this approach are then considered to develop a more general CS-based weather radar processing algorithms in presence of noise, ground clutter and dual-polarimetric data. Finally, a super-resolution approach is presented for the spectral recovery of an undersampled signal when certain frequency information is known.