We present a historical review of the theory of the Zeeman effect in molecular lines, from its discovery at the end of the 19th century till today. The fast development of the quantum theory owes much to the impulse given by the experimental investigation of the molecular Zeeman effect. Laboratory experiments produced fruitful results after the predictions made by the quantum theory. The introduction by Racah of the powerful angular momentum algebra gave a second impulse to the theory and allowed to fully understand the fine structure and polarization properties of molecular transitions. At present, the theory of the Zeeman and Paschen-Back effects in molecular lines is being applied to spectro-(polarimetric) observations to infer the physical and magnetic properties of cold plasmas in the atmospheres of the Sun and of other stars.