A Moebius transformation is a function of a complex variable, defined on an extended complex plane. It is a conformal transformation, meaning that it preserve angles and orientation. The inverse function of a Moebius transformation is also a Moebius transformation. Some special cases of Moebius transformations are well known geometric transformations: translation, rotation, reflection, homothety and inversion, which are described in this work. Before constructing an inverse transformation, it is important to mention the construction of a symmetric point with respect to the unit circle as a part of constructing an inversion. Any Moebius transformation is a composition of translation, rotation, reflection, homothety and inversion. We give a proof that the Moebius transformation maps lines and circles into lines or circles, where we consider a line as a circle with radius equal to infinity. Before the proof of the implicit formula for the mapping, which is used for mapping three distinct points onto three distinct points, we prove that non-identity Mobius transformations have two fixed points. We give examples using the implicit formula and visualize them for better understanding. For finding a transformation witch maps a given area into another given area we apply the symmetry-preserving property of Mobius transformations. That property is proven and used in examples. The Moebius transformation is very useful and has applications in diferent science fields, based on the mapping the Riemman Sphere to a flat plane. Some of those applications are described here.