Aeroelasticity studies the static and dynamic interaction between structural deformation and fluid forces. As a result, the aeroelasticity is usually divided into three parts: aerodynamics, structural response, and dynamics with statics as a special case. Instabilities may occur to this interaction (feedback) that lead to structural failure and, even when no instability occurs, the interaction may lead to degradation or improvement of the system performance. There are several unstable phenomena may occur for elastic bodies such as flutter, divergence, low cycle oscillation, buffet, and control surface reversal. These unstable phenomena can be classified as dynamic or static. The present work provides a tutorial for those newly encountering aeroelasticity and a review of the recent literature from this century (after 2000). This includes mathematical modelling and its applications to airplanes, rotor blades, energy harvesting, and the control, and optimization of aeroelastic systems. Recent research advances are summarized and some suggestions for future work are made.