© 2019, This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply. Light’s capacity to carry angular momentum is integral to our knowledge of physics and ability to probe matter. In addition to spin, photons can occupy free-space orbital angular momentum eigenstates 1,2 . Visible light orbital angular momentum is used in quantum information experiments, super-resolution microscopy, optical tweezers and angular momentum transfer to atoms in optical lattices 3,4 . Soft X-ray orbital angular momentum applications, slowed by the lack of suitable optics and the rarity of coherent X-ray sources, could enable the direct alteration of atomic states through orbital angular momentum exchange, and methods to study the electronic properties of quantum materials. We have made soft X-ray diffractive optics that generate single Laguerre–Gauss modes, observed carrying up to 30ħ angular momentum per photon, or their superpositions. We also present Hermite–Gauss diffractive optics and a soft X-ray orbital angular momentum analyser. These tools could enable both the manipulation and finer characterization of topologically complex electronic matter, such as magnetic skyrmions.