Free-electron lasers and high-harmonic-generation table-top systems are new sources of extreme-ultraviolet to hard X-ray photons, providing ultrashort pulses that are intense, coherent and tunable. They are enabling a broad range of nonlinear optical and spectroscopic methods at short wavelengths, similar to those developed in the terahertz to ultraviolet regimes over the past 60 years. The extreme-ultraviolet to X-ray wavelengths access core transitions that can provide element and orbital selectivity, structural resolution down to the sub-nanometre scale and, for some methods, high momentum transfers across typical Brillouin zones; the possibilities for polarization control and sub-femtosecond time resolution are opening up new frontiers in research. In this Roadmap, we review the emergence of this field over the past 10 years or so, covering methods such as sum or difference frequency generation and second-harmonic generation, two-photon absorption, stimulated emission or Raman spectroscopy and transient grating spectroscopy. We then discuss the unique opportunities provided by these techniques for probing elementary dynamics in a wide variety of systems.

X-ray free-electron lasers and high-harmonic-generation sources of extreme-ultraviolet (EUV) to hard X-ray photons deliver intense ultrashort pulses and enable the extension of nonlinear methods to much shorter wavelengths.EUV to X-ray wavelengths access core transitions that can provide element and orbital selectivity. These wavelengths also achieve sub-nanometre structural resolution and high momentum transfer, with femtosecond and attosecond time resolution.Nonlinear EUV/X-ray methods that have emerged include sum or difference frequency generation, parametric down-conversion, second-harmonic generation, two-photon absorption, stimulated emission or Raman spectroscopy and transient grating spectroscopy.Nonlinear EUV/X-ray science is developing hand-in-hand with instrumentation, to improve pulse features and enhance accessibility with the use of table-top systems or compact accelerators.These techniques offer unique opportunities for probing dynamical events in a wide variety of systems, including surface and interface processes, chirality, nanoscale transport and multidimensional core-level spectroscopy.

New sources of extreme-ultraviolet to hard X-ray photons have enabled a wide range of short-wavelength nonlinear optical and spectroscopic methods over the past decade, and, for the future, offer unique opportunities to probe elementary dynamics in various systems.