Diffraction Gratings for X-ray Astronomy

We briefly review the physical principles of diffraction grating spectrometers for astrophysical X-ray spectroscopy, in the context of the resolution and sensitivity requirements for a wide range of spectroscopic diagnostics. We give a brief description of the manufacturing techniques that gave us the gratings on the Chandra and XMM-Newton observatories. We provide some examples of astrophysical spectroscopy with these instruments. We describe the instrumental innovations that grew out of these projects, which are baselined as the diffraction elements for the currently planned next generation of X-ray grating spectrometers.