[en] We formulate the Dirac equation for a massive neutral spin-half particle on a rotating black hole spacetime, and we consider its (quasi)bound states: gravitationally-trapped modes which are regular across the future event horizon. These bound states decay with time, due to the absence of superradiance in the (single-particle) Dirac field. We introduce a practical method for computing the spectrum of energy levels and decay rates, and we compare our numerical results with known asymptotic results in the small-$M\mathrm{\mu <!--\; ??\; -->}$ and large-$M\mathrm{\mu <!--\; ??\; -->}$ regimes. By applying perturbation theory in a horizon-penetrating coordinate system, we compute the ‘fine structure’ of the energy spectrum and find good agreement with numerical results. We obtain data for a hyperfine splitting due to black hole rotation. We evolve generic initial data in the time domain, and show how Dirac bound states appear as spectral lines in the power spectra. In the rapidly-rotating regime, we find that the decay of low-frequency co-rotating modes is suppressed in the (bosonic) superradiant regime. We conclude with a discussion of physical implications and avenues for further work. (paper)