Soft-condensed matter physics has provided, in the past decades, many of the relevant concepts and methods allowing successful description of living cells and biological tissues. This recent quantitative physical description of biological systems has profoundly advanced our understanding of life, which is shifting from a descriptive to a predictive level. Like other active materials investigated in condensed matter physics, biological materials still pose great challenges to modern physics as they form a specific class of nonequilibrium systems. Actively driven membranes have been studied for more than two decades, taking advantage of rapid progress in membrane physics and in the experimental development of reconstituted active membranes. The physical description of activity within living biological membranes remains, however, a key challenge that animates a dynamic research community, bringing together physicists and biologists. Here, we first review the past two decades of experimental and theoretical advances that enabled the characterization of mechanical properties and nonequilibrium fluctuations in active membranes. We distinguish active processes originating from membrane proteins or from external interactions, such as cytoskeletal forces. Then, we focus on the emblematic case of red blood cell flickering, the active origin of which has been debated for decades until recently. We finally close this review by discussing future challenges in this ever more interdisciplinary field.