  X-ray echo spectroscopy, a counterpart of neutron spin-echo, is being
introduced here to overcome limitations in spectral resolution and weak signals
of the traditional inelastic x-ray scattering (IXS) probes. An image of a
point-like x-ray source is defocused by a dispersing system comprised of
asymmetrically cut specially arranged Bragg diffracting crystals. The defocused
image is refocused into a point (echo) in a time-reversal dispersing system. If
the defocused beam is inelastically scattered from a sample, the echo signal
acquires a spatial distribution, which is a map of the inelastic scattering
spectrum. The spectral resolution of the echo spectroscopy does not rely on the
monochromaticity of the x-rays, ensuring strong signals along with a very high
spectral resolution. Particular schemes of x-ray echo spectrometers for
0.1--0.02-meV ultra-high-resolution IXS applications (resolving power $> 10^8$)
with broadband $\simeq$~5--13~meV dispersing systems are introduced featuring
more than $10^3$ signal enhancement. The technique is general, applicable in
different photon frequency domains.
