Historically, one distinguishes between the normal and an anomalous Zeeman effect (discovered by Thomas Preston in Dublin, Ireland). The anomalous effect appears on transitions where the net spin of the electrons is non-zero. It was called "anomalous" because the electron spin had not yet been discovered, and so there was no good explanation for it at the time that Zeeman observed the effect. Wolfgang Pauli recalled that when asked by a colleague as to why he looked unhappy, he replied, "How can one look happy when he is thinking about the anomalous Zeeman effect?"
At higher magnetic field strength the effect ceases to be linear. At even higher field strengths, comparable to the strength of the atom's internal field, the electron coupling is disturbed and the spectral lines rearrange. This is called the Paschen–Back effect.
In the modern scientific literature, these terms are rarely used, with a tendency to use just the "Zeeman effect".
Another rarely used obscure term is inverse Zeeman effect, referring to the Zeeman effect in an absorption spectral line.
A similar effect, splitting of the nuclear energy levels in the presence of a magnetic field, is referred to as the nuclear Zeeman effect.