In this Book we have presented a number of different new classes of exactly solvable quantum systems in non relativistic quantum mechanics applying a scheme called the Extended transformation (ET) method, in any arbitrary D-dimensional Euclidean spaces, within the framework of Green’s function technique. The procedure to apply ET is to select a potential term(s) of the original potential termed as the “working potential”. The working potential eventually specifies the form of the basic transformation function, with the use of which the ET method is implemented. In case of multi-term potential the transformation procedure may be applied repeatedly, by selecting working potential differently to generate a variety of solved quantum system, the number of solvable quantum system in principle, being equal to the number of ways (2^n-1). A major complication that always arise dealing with non-power low potential is that the newly generated QSs are always of a Sturmian type. It behaves like a system index enumerating different QSs having a single bound state. As no standard procedure to convert Sturmian QSs to normal QSs exists, sometimes we have to use case specific regrouping techniques.