The paradigm of Compressive Sensing (CS) has emerged in the last few years as one of the most powerful and flexible tools to solve challenging electromagnetic design, retrieval, and inverse problems. This success is motivated by the capability of CS methodologies and algorithms to reliably and efficiently address sampling problems (i.e., finding the minimum number of non-adaptive samples able to fully describe a target phenomenon) and ill-posed recovery problems (i.e., identifying a certain signal/phenomenon starting from a reduced set of measurements), as well as by their capability to overcome the classical limitations enforced by Nyquist theorem. In fact, such efficiency and effectiveness is demonstrated by several successful CS applications in microwave imaging, antenna diagnostics, and array design. The objective of this work is to give a broad review of the fundamentals of CS in Electromagnetics, as well as to illustrate the most important open challenges and future trends.