Since the theoretical aeroelasticity for flapping-wing aerodynamics was introduced in the 1920s, the effects of flexibility on aeroelasticity have been paid more attention to aerodynamic design. In recent years, the trait of the wing flexibility is applied for small-scale wings of biomimetic flyers including micro air vehicles and mini unmanned aerial vehicles. Until now, the aerodynamic performance and great agility of these flyers, which are beneficially used for military missions and other civilian applications, have been improved through surrogate flapping wings with the favorable effects of the flexibility. As per the aeroelasticity principle for the forward flying, the chordwise flexibility of an elastic flapping wing can generate thrust and lift redistributions, whereas the spanwise flexibility can result in variations of the angle of attack and the shift of phase along the wingspan direction. Consequently, all vortices generated by the flapping wing i.e. (1) leading-edge vortices, (2) tip vortices, and (3) trailing-edge vortices are blended supportively, thereby improving the aerodynamic performance and agility. Hence, the growth of research and development of the aerodynamic performance and agility for these flyers under the influence of flexible wings increases through experimental and computational studies dynamically and rapidly. This review aims to highlight the important role of the flexibility in the recent progress in wing aerodynamics of these flyers through several wing models done by famous groups of experts in this field. In addition, this review includes the acoustics of the wings under the flexibility effects which is considered as a new key for better flyer design and improvement. A comprehensive understanding of the integrated aerodynamics and acoustics under the wing flexibility is, therefore, needed.