In this study, a cubic NURBS-based isogeometric finite element method is developed to study the aeroelastic tailoring of variable stiffness composite laminated (VSCL) plates, including skew, skew-tapered, tapered, and circular shapes based on third-order shear deformation theory. Also, the first-order piston theory is employed to approximate aerodynamic loads with arbitrary yaw flow angles over the plate area to establish aeroelastic equations. The NURBS basis functions are then utilized to simulate exact geometries and estimate variable fields for discretizing governing equations by well-known isogeometric analysis. Finally, the free vibration and flutter analysis of a VSCL quadrilateral and circular plate will be presented via the obtained eigenvalue problem. The achieved results are compared with the available results in the literature to prove the accuracy and validity of the present method. The effect of different parameters such as the flow yaw angle, end and middle-length fiber orientation, longitudinal to transverse Young’s modulus ratio, geometric shape, boundary conditions, and turning symmetric layup to asymmetric layup is carried out on the critical flutter aerodynamic pressure and natural frequencies.