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For many years conformal transformations have been studied. Riemann surfaces of many of them have been constructed and their principal properties have been discovered and examined in the study of the theory of functions of a complex variable, for if a transformation is conformal it must satisfy the Cauchy-Riemann equations as will be shown later. Comparatively little, however, has ever been done with conformal transformations. In this paper, we have made a study of some of the properties of such transformations and have constructed Riemann surfaces of a few of them. In Part I, an ordinary conformal transformation has been considered and its Riemann surface constructed. Then by means of non-conformal transformations upon the conformal one, non-conformal transformations between real variables were obtained, and through the properties of the former, the properties of the latter were discovered and the Riemann surfaces were constructed. In Part II, the non-conformal transformations were studied directly.