In this article the design, optimization and characterization of diffractive optical
elements formed on a curved surface are reviewed. For such curved diffractive optical
elements not only the phase function, but also the surface shape are free parameters
that can be used for optimization, yielding much better performances than both flat
diffractive optical elements and reflective/refractive optical elements when operating
with quasimonochromatic light. We present a new analytic design approach for the surface
shape that ensures uniform collimation of a light source with any angular distribution.
We demonstrate the usefulness of this design also for ideal (brightness conserving)
collimation and concentration of diffuse light, aberration-free imaging, and optical
Fourier transform. We present experimental results that confirm our theoretical analysis.