The main purpose of this work is to calculate the isotropic shielding of some nuclei, other than protons, of first- and second-row atoms in the periodic table and some transition metal elements in a wide variety of molecular environments, with a view to gaining an understanding of the various electronic factors which determine the observed nuclear shielding. Chapter One introduces some general concepts. Chapter Two presents a general servey of various Semi-empirical molecular orbital methods. Various MO calculations of nuclear shielding are briefly reviewed in Chapter Three with particular emphasis on Pople's GIAO-MO approach. This chapter also contains a description of the theory of medium effects on nuclear shieldings. Chapter Four is concerned with approximate MO theories and their application to the GIAO-MO method for Sum-Over-State (SOS) results of some shieldings of first-row elements. The medium effect models are used to explore the role of medium effects on nuclear shielding. This exploration is supplemented by performing some calculations on hydrogen-bonded models. Chapter Five records the SOS results for the nuclear shielding of some second-row elements in conjunction with the GIAO-MO method. The solvaton model is also used in an attempt to improve on our understanding of the relation between magnetic shielding and various features of molecular electronic structure. Chapter Six presents some shielding data for transition metal elements obtained by means of Pople's GIAO-MO method in conjunction with the INDO/5R parameterization scheme for SOS results of some inorganic molecules. A reasonable correlation between the calculated and experimental chemical shifts, compared by mean of a least-squares fit is obtained for some of the molecules considered.