Mass loss from massive stars plays a determining role in their evolution through the upper Hertzsprung-Russell diagram. The hydrodynamic theory that describes their steady-state winds is the line-driven wind theory (m-CAK). From this theory, the mass loss rate and the velocity profile of the wind can be derived, and estimating these properly will have a profound impact on quantitative spectroscopy analyses from the spectra of these objects. Currently, the so-called b law, which is an approximation for the fast solution, is widely used instead of m-CAK hydrodynamics, and when the derived value is beta greater than or similar to 1.2, there is no hydrodynamic justification for these values. This review focuses on (1) a detailed topological analysis of the equation of motion (EoM), (2) solving the EoM numerically for all three different (fast and two slow) wind solutions, (3) deriving analytical approximations for the velocity profile via the LambertW function and (4) presenting a discussion of the applicability of the slow solutions.