With the construction of the European Spallation Source, a remarkable opportunity has emerged to conduct high sensitivity searches for neutron oscillations, including a first search for thirty years for free neutrons converting to antineutrons. Furthermore, searches can be made for transitions of neutrons and antineutrons to sterile neutron states. The HIBEAM program provides an increase in sensitivity of an order of magnitude compared to previous work. The HIBEAM program corresponds to baryon number violation by one and two units. The observation of a process satisfying a Sakharov condition addresses the open question of the origin of the matter-antimatter asymmetry in the Universe. Sterile neutron states would belong to a `dark' sector of particles which may explain dark matter. As electrically neutral, meta-stable objects that can be copiously produced and studied, neutrons represent an attractive portal to a `dark' sector. This paper describes the capability, design, infrastructure, and potential of the HIBEAM program. This includes a dedicated beamline, neutron optical system, magnetic shielding and control, and detectors for neutrons and antineutrons.