The problem of the origin of the Earth magnetic field is introduced and discussed in mathematical terms. The first instability relevant to this problem is the onset of thermal convection. The latest theoretical developments are presented. These involve the important distinction between local and global instabilities. The onset of dynamo action is then addressed. Particular attention is given to the concepts of weak and strong field and the relevant bifurcation sequence. Comparison of asymptotic and numerical models is performed, and mixed asymptotic and numerical models are introduced. The challenging limiting case of vanishing viscosity and inertia, known as the ‘Taylor state’, is then introduced. Deviations in terms of viscous ‘Ekman state’ or inertial ‘torsionnal oscillations’ are presented. Finally, scaling laws, which are sometimes used as a tool to relate numerical simulations to the parameter regime relevant to the Earth’s core, are presented and discussed.