Sparsity is an attribute present in a myriad of natural signals and systems, occurring either inherently or after a suitable projection. Such signals with lots of zeros possess minimal degrees of freedom and are thus attractive from an implementation perspective in wireless networks. While sparsity has appeared for decades in various mathematical fields, the emergence of compressed sensing (CS) - the joint sampling and compression paradigm - in 2006 gave rise to plethora of novel communication designs that can efficiently exploit sparsity. In this monograph, we review several CS frameworks where sparsity is exploited to improve the quality of signal reconstruction/detection while reducing the use of radio and energy resources by decreasing, e.g., the sampling rate, transmission rate, and number of computations. The first part focuses on several advanced CS signal reconstruction techniques along with wireless applications. The second part deals with efficient data gathering and lossy compression techniques in wireless sensor networks. Finally, the third part addresses CS-driven designs for spectrum sensing and multi-user detection for cognitive and wireless communications.