A dynamo driven by motions unaffected by viscous forces is termed magnetostrophic. Although such a model might describe well magnetic field generation in Earth’s core, its existence is in doubt as numerical simulators have to impose substantial viscosity to stabilize solutions of the full MHD dynamo equations. An attempt is made here to revive interest in a procedure proposed by Taylor [Proc. R. Soc. Lond. A, 1963, 274, 274] for finding inertialess magnetostrophic dynamos. The evolution of the magnetic field from the fluid flow follows the usual kinematic path, but the creation of the zero viscosity flow from the magnetic field was reduced by Taylor to the solution of a second-order ordinary differential equation. Roberts and Wu [Geophys. Astrophys. Fluid Dyn., 2014, 108] derived an exact solution of this equation for axisymmetric mean-field dynamos. Numerical solutions of this equation are presented here, leading to the first truly magnetostrophic dynamos ever found. The magnetic field and fluid flow are derived and discussed for - and -dynamos.