Dirac's groundbreaking 1931 theory of magnetic monopoles made it possible to consider such elementary particles within the contexts of classical electrodynamics and quantum mechanics. Despite years of searching, no magnetic monopole has been convincingly identified experimentally. Nevertheless, one can perform experiments in systems that permit the existence of close analogues of these elusive particles. One such analogue is the Dirac monopole that we recently created and observed in the synthetic magnetic field generated by a spinor Bose-Einstein condensate. The response of the condensate to the presence of a monopole reveals the characteristic features of the system envisioned by Dirac. In related experiments, we create and detect quantum point defects within the condensate that are analogues of 't Hooft-Polyakov magnetic monopoles, which are expected to arise as a result of symmetry breaking in grand unified field theories.