We present three-dimensional numerical magnetohydrodynamic simulations of radiatively inefficient spherical accretion onto a black hole. The simulations are initialized with a Bondi flow, and with a weak, dynamically unimportant, large-scale magnetic field. The magnetic field is amplified as the gas flows in. When the magnetic pressure approaches equipartition with the gas pressure, the field begins to reconnect and the gas is heated up. The heated gas is buoyant and moves outward, causing line stretching of the frozen-in magnetic field. This leads to further reconnection, and more heating and buoyancy-induced motions, so that the flow makes a transition to a state of self-sustained convection. The radial structure of the flow changes dramatically from its initial Bondi profile, and the mass accretion rate onto the black hole decreases significantly. Motivated by the numerical results, we develop a simplified analytical model of a radiatively inefficient spherical flow in which convective transport of energy to large radii plays an important role. In this "convection-dominated Bondi flow" the accretion velocity is highly subsonic and the density varies with radius as ~R^{-1/2} rather than the standard Bondi scaling ~R^{-3/2}. We estimate that the mass accretion rate onto the black hole is significantly less than the Bondi accretion rate. Convection-dominated Bondi flows may be relevant for understanding many astrophysical phenomena, e.g. post-supernova fallback and radiatively inefficient accretion onto supermassive black holes, stellar-mass black holes and neutron stars.

23 pages, 6 figures, submitted to ApJ