The measurement of natural circular dichroism has been extended to the X-ray region. XNCD spectra have been measured for uniaxial single crystals of the chiral coordination compounds Na-3[Nd(digly)(3)]. 2NaBF(4). 6H(2)O (digly = 2,2'-oxydiacetate) and {[Co(en)(3)]Cl-3}(2). NaCl . 6H(2)O (en = ethane-1,2-diamine) and for ionic crystals such as LiIO3. The XANES part of the XNCD shows circular dichroism corresponding to chiral multiple scattering paths of the photoelectron. In addition, both the Nd(III) and Co(III) compounds show quadrupole allowed pre-edge features (2p --> 4f for Nd and Is --> 3d for Co) which have exceptionally large Kuhn dissymmetry factors. The theory of XNCD is discussed and the circular dichroism is shown to be due to the interference between allowed electric dipole and electric quadrupole transition moments (E1-E2 mechanism). This new spectroscopy represents a useful extension to techniques for the study of both molecular and crystal structural enantiomorphism.