Leaky-wave antenna (LWA), as a special kind of waveguiding structure simultaneously characterized by both the guidance and radiation of electromagnetic waves along it, has received much attention and has been in the spotlight of both academia and industry since its inception in the 1940s. This is mainly because LWAs possess many appealing properties such as the highly directive beam, easy feeding mechanism, simple configuration, and particular frequency-driven beam-scanning, which enable them to hold promise in the development of a unique front-end antenna solution than others (e.g., reflector antenna and phased array). In addition to these inherent features, the exploitation of LWAs featuring multifarious sophisticated electrical functionalities (i.e., functional LWAs), with resort to engineering relevant radiating discontinuities and waveguiding structures, constitutes one mainstream research in the communities and increasingly prevails in the research & development of modern multifunctional wireless RF/microwave systems recently. This article is structured under this background, which is mainly dedicated to reviewing and investigating several enabling concepts, structures, and techniques that are specifically applied to the design of functional LWAs characterized by, for example, simultaneously low side-lobe and cross-polarization behaviors, rapid beam-scanning, stable radiation, frequency-selectivity, and open-stopband suppression. To make the technical discussions much more self-contained and systematic, this article begins with leaky-wave basics as well as general geometrical and electrical characteristics of LWAs. In addition, some representative system-level applications enabled by LWAs are briefly described, together with several promising research directions that can be further explored and galvanized, from the authors' perspective.