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.