Traditional Fresnel zone plates (TFZPs) can be used for the focusing and imaging of soft X-rays and extreme ultraviolet (EUV) radiations. However, the focal spot size of a TFZP is limited in technology by the minumum feature size (20-40 nm) that can be fabricated by lithography. Recently, a novel diffractive optical element called a photon sieve, which consists of a great number of pinholes properly distributed over the Fresnel zone, was proposed to overcome the resolution problem of a TFZP. Following the initial Nature paper, we have presented the simple individual far-field model for photon sieves and established the general theory for various modified FZPs. Here we present an overview of our research work in these areas. The related contents include the analytical descriptions for the diffracted fields of the individual pinholes and of the individual open rings, the selection conditions for the pinholes and for the open rings, the suppression of sidelobes, the physical limit of the resolution, the suppression of higher-order foci, the construction of a specific focal spot shape like a Gaussian focal spot, the equivalent pupil (or aperture) function theory, the diffraction efficiency, and the fast computation method for the individual diffracted fields. We also discuss the extension of the equivalent pupil function theory to apodized multilevel diffractive lenses.