The quantized orbital angular momentum (OAM) of photons1 offers an additional degree of freedom and topological protection from noise. Photonic OAM states have therefore been exploited in various applications(2,3) ranging from studies of quantum entanglement and quantuminformation science(4-7) to imaging(8-12). TheOAM states of electron beams(13-15) have been shown to be similarly useful, for example in rotating nanoparticles and determining the chirality of crystals(16-19). However, although neutrons-as massive, penetrating and neutral particles-are important in materials characterization, quantum information and studies of the foundations of quantum mechanics, OAM control of neutrons has yet to be achieved. Here, we demonstrate OAM control of neutrons using macroscopic spiral phase plates that apply a ` twist' to an input neutron beam. The twisted neutron beams are analysed with neutron interferometry. Our techniques, applied to spatially incoherent beams, demonstrate both the addition of quantum angular momenta along the direction of propagation, effected by multiple spiral phase plates, and the conservation of topological charge with respect to uniform phase fluctuations. Neutron-based studies of quantum information science(20,21), the foundations of quantum mechanics(22,23), and scattering and imaging(24) of magnetic, superconducting and chiral materials have until now been limited to three degrees of freedom: spin, path and energy. The optimization of OAMcontrol, leading to well defined values ofOAM, would provide an additional quantized degree of freedom for such studies.