  In this paper, we discuss the concept of $\rho$-regularity of analytic map
germs and its close relationship with the existence of locally trivial smooth
fibrations, known as the Milnor fibrations. The presence of a Thom regular
stratification or the Milnor condition (b) at the origin, indicates the
transversality of the fibers of the map G with respect to the levels of a
function $\rho$, which guarantees $\rho$-regularity. Consequently, both
conditions are crucial for the presence of open book structures and the Milnor
fibrations. The work aims to provide a comprehensive overview of the main
results concerning the existence of Thom regular stratifications and the Milnor
condition (b) for germs of analytic maps. It presents strategies and criteria
to identify and ensure these regularity conditions and discusses situations
where they may not be satisfied. The goal is to understand the presence and
limitations of these conditions in various contexts.
