We introduce a new class of reflectarrays, namely, the aperiodic conformal reflectarrays, aimed at exploiting, as much as possible, the available degrees of freedom of the radiating structure, such as positions, orientations and characteristics of the radiating elements and the shape of the reflecting surface. A synthesis technique is outlined, properly dealing with key aspects such as the complexity of the employed radiative model, the algorithms exploited to achieve the final solution, the hardware adopted to run the synthesis code, and the constraints dictated by physical and manufacturing requirements. The potentialities of aperiodic conformal reflectarrays are discussed, also with the aid of a numerical example.