Inspired by the anti-reflection functionality of cicada wings decorated with nanocone arrays, a facile technique to endow flexible polymer substrates of diverse chemical compositions with the same functionalities has been devised. In this universally applicable two-step technique based on a capacitatively coupled radiofrequency plasma (CCRP), first oxygen-plasma treatment is implemented to grow arrays of vertical elongated nanostructures with almost uniform cross-sectional diameter of the polymer substrate, and then fluorocarbon polymer deposition is carried out so that the nanostructures evolve into nanocones with small apex angle. The dependence of ion-bombardment-induced sputtering on the local impingement angle of ions taper the vertical nanostructures into the nanocones during the fluorocarbon polymer deposition under CCRP. The nanocone arrays exhibit low specular reflectance in a broad wavelength range and a wide incidence angle range that is quite insensitive to the polarization state of the incident light. The effective refractive index gradient of irregularly arranged nanocone arrays is characterized from the refractive index of the fluorocarbon polymer and the volume fraction considering the nanocone probability with the Gaussian distribution. The excellent broadband and omnidirectional anti-reflection properties are in consequence of a graded refractive index.