In this review, we describe our method for creating holograms of incoherent objects, dubbed Fresnel incoherent correlation holography (FINCH). FINCH creates holograms by a single-channel on-axis incoherent interferometer process. Like any Fresnel hologram, the object is correlated with quadratic phase functions, but the correlation is carried out without any movement. Generally, in the FINCH system, light is reflected, or emitted, from a three-dimensional (3D) object, propagates through a spatial light modulator (SLM), and is recorded by a digital camera. The SLM is used as a beam-splitter of the single-channel incoherent interferometer, such that each spherical beam originated from each object point is split into two spherical beams with two different curve radiuses. Incoherent summing of the entire interferences between all the couples of the spherical beams creates the Fresnel hologram of the observed 3D object. When this hologram is reconstructed in the computer, the 3D properties of the object are revealed. In this review, we describe various aspects of FINCH which have been described recently, including FINCH of reflected white light, FINCH of fluorescence objects, a FINCH-based holographic fluorescence microscope, a FINCH configuration which capitalizes on the polarization sensitivity of the SLM and finally FINCH is analyzed in view of linear system theory.