Inhomogeneous magnetic fields exert a magnetic body force on electrically nonconducting, magnetically permeable fluids subject to a thermal gradient. This force can be utilized to compensate for gravity and thereby to control convection. The field effect on convection is represented by a dimensionless vector parameter \rmR_m=\mu_0\alpha\chi_0d^3\Delta Tøver\rho_0\nu D_T \left(\rmH\cdot\nabla\rmH\right)_\rmr=0^ext for diamagnetic fluids or \rmR_m=\mu_0\chi_0d^3\Delta Tøver\rho_0T_0\nu D_T\left( \rmH\cdot\nabla\rmH\right)_\rmr=0^ext for paramagnetic fluids. This parameter measures the strength of the induced magnetic buoyancy force due to the applied field gradient relative to the dissipative effects. Its vertical component competes with gravitational buoyancy, and a critical relationship between this component and the usual Rayleigh number is identified for the onset of convection. Predictions for paramagnetic fluids agree with the experiments. Magnetically driven convection should be observable even in diamagnetic fluids such as pure water using current magnetic technology.