This paper reviews the development of body-fitted coordinate systems for a wide variety of two-dimensional flows. The author classifies the techniques used in the body fitted transformation as: (1) algebriaic techniques ( simple stretching and shearing), (2) partial differential equation techniques, and (3) conformal mapping techniques. The conformal mapping is widely used in fluid Mechanics. It automatically stresses the regions of interest near the leading and trailing edges of airfoils. In addition, conformal mapping simplifies the kinematics aspect of the fluid mechanics problem. For these reasons, a new modified conformal mapping technique is initiatedand presented in the second part of this paper. In appendix, a computer code based on this technique is listed and some few results are plotted.