Structural studies of two-dimensionally ordered phases made of bent-core mesogens are presented, with the emphasis on the X-ray studies of the columnar phases built of smectic layer fragments. We present the evidence for two types of packing of layer fragments: the B1-type structures where the layers are broken into blocks arranged into the orthogonal body centered lattice and the B1rev-type structures made of slightly deformed layers, where the primitive crystallographic unit cell is in general oblique. In some cases two-dimensional structures made of bilayers are observed. Bilayers are characteristic for the general tilt phases, which possess a longitudinal polarization component, in the direction perpendicular to the smectic layer fragment. We also outline the theoretical model predicting and describing the observed structures and present a model which enables the calculation of the electron density maps from the X-ray diffraction data.