An X-ray source for producing soft X-rays, the X-ray source comprising: a cathode having an electron-emitting structure supported by a support structure, the electron-emitting structure being at least partially transparent to X-rays within a region bounded by the support structure； an anode having an X-ray emitting surface parallel to the electron-emitting structure of the cathode； and an electrically insulating spacer arranged between the anode and the cathode； wherein the electron-emitting structure of the cathode and the X-ray emitting surface of the anode are arranged such that, in use, the electron-emitting structure is operable to bombard the anode with electrons, causing X-rays to be emitted from the X-ray emitting surface and to pass through the cathode； and wherein the insulating spacer is arranged between the anode and the support structure of the cathode and projects beyond the support structure, across part of the anode, into the said region.
