We have discussed the main application areas of diffractive optical elements fabricated on a curved substrate. The versatility of the CDOE comes from the fact that the substrate shape and the grating function can be controlled independently. CDOEs can be used to achieve aberration-free imaging, uniform collimation and ideal concentration of diffuse light at the thermodynamic limit, for general source and target shapes. In high-NA focusing CDOEs can be used to control the effective apodization factor of the focused light rays, and thereby yield uniform and tight focal spots with very low sidelobe intensities. For many applications - like the ideal concentration of diffuse light discussed in Sections 3 and 4, the vectorial diffraction problems mentioned in Section 5, and the optical coordinate transformation described in Section 7 - it is sufficient to optimize the shape of the CDOE, and the grating function can be recorded either holographically with simple spherical or plane waves, or by direct laser lithography. In other cases the grating function must have a more complicated form; here additional optics or computer-generated holograms may be required. Since CDOEs are diffractive elements, they usually have large chromatic aberrations which limit their use to quasi-monochromatic light, except for the spectrograph applications discussed in Section 6. The history of CDOEs began in the mid-19th century with Rowland's curved spectroscopic grating. Thanks to their versatility, and to recent advances in applications, in new design methods, and in holographic recording materials that can be placed on arbitrary shapes and enable high diffraction efficiency, CDOEs should remain a lively area of research in the future too.