The main aim of the present work has been to present some theoretical results of nuclear magnetic shieldings for some first row nuclei in a variety of different electronic environments, in order to obtain an improved description of the various factors contributing to nuclear shielding. In Chapter One, several current theories of magnetic shielding are briefly reviewed for purposes of comparison. Chapter Two presents a general survey of various semi-empirical molecular orbital methods of magnetic shielding with particular emphasis on Pople's GIAO-MD approach. This chapter also contains a description of the theory of medium effects on nuclear shielding and a brief introduction to the solvaton model. Chapters Three to Six present 11B, 13C, 14N (or 15N), 17O and 19F chemical shifts, shielding constants and their anisotropies calculated by means of Pople's GIAO-MO method in conjunction with the INDO/S and MINDO/3 parameterization schemes for a wide variety of organic and inorganic molecules. It is demonstrated that the results of the INDO/S calculations are in reasonable agreement with the experimental data, whereas the MINDO/3 parameterization scheme provides quantitatively unreliable results for all of the nuclei considered here. The results obtained are analysed in an attempt to improve our understanding of the relation between magnetic shielding and various features of molecular electronic structure. Finally, the solvaton model is used to explore the role of polar effects on 13C, 14N (or 15N), 17O and 19F shieldings of some polar compounds. This exploration is supplemented by performing some hydrogen-bonding calculations for some model compounds.