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Together with relativity theory, quantum mechanics stands as the conceptual foundation of modern physics - forming the basis by which we understand the minute workings of the subatomic world. But at its core lies a paradox; standard conceptions of quantum mechanics imply that many of the actual measurements whose results we take to support and verify quantum mechanical theory, can have no definite outcomes. Some quantity such as position or momentum is always indefinite on a quantum system; and if an indefinate quantity is measured, the macroscopic state of the measuring apparatus that is supposed to record the outcome instead becomes indefinite itself. This text offers insights from a range of leading physicists into this issue.