The flexoelectricity describes the contribution of the linear couplings between the electric polarization and strain gradient and between polarization gradient and strain to the thermodynamics of a solid and represents the amount of polarization change of a solid arising from a strain gradient. Although the magnitude of the flexoelectric effect is generally small, its contribution to the overall thermodynamics of a solid may become significant or even dominant at the nanometer scale. Recent experimental and computational efforts have led to significant advances in our understanding of the flexoelectric effect and its exploration of potential applications in devices such as sensors, actuators, energy harvesters, and nanoelectronics. Here we review the theoretical development and experimental progress in flexoelectricity including the types of materials systems that have been explored and their potential applications. We discuss the challenges in the experimental measurements and density functional theory computations of the flexoelectric coefficients including understanding the order of magnitude discrepancies between existing experimentally measured and computed values. Finally, we offer a perspective on the future directions for research on flexoelectricity.