Interpreting a scientific concept (i.e., identifying or constructing it in any particular instance) is a complex cognitive task. We analyze the underlying knowledge required to achieve such concept interpretation accurately and efficiently. This analysis is then used to examine our detailed observations of expert scientists and novice students interpreting the physics concept "acceleration." Most of the experts interpret the concept well in expected ways; however, even some experienced scientists exhibit marked deficiencies in concept interpretation. Novice students, in spite of using a scientific concept for some months, interpret it incorrectly in many cases; their poor performance can be traced to concept knowledge which consists largely of disconnected knowledge fragments, and which can often not be properly applied even if correctly stated. Instructional experiments, which directly address such deficiencies, indicate that it is possible to improve substantially students' ability to interpret a scientific concept.