Milnor Fibrations and the Concept of d-regularity for Analytic Map Germs

In this expository article we review Milnor's proof of his celebrated fibration theorem and the way how it extends to the case of real analytic map germs. We highlight the similarities and the differences between the complex analytic and the real analytic settings, and we indicate the key geometric aspects in Milnor's proof that make it work in the complex case, unlike the theorem in the real setting which is more stringent. We explain too how, from the viewpoint of Milnor type fibrations, the difference between the real and the complex settings naturally leads to the concept of d-regularity.