The main object of the present work has been to calculate nuclear magnetic shieldings for [11]B, [13]C, [14]N (or [15]N), [17]O and [19]F nuclei in a wide variety of organic and inorganic molecules with a view to the understanding of the various electronic factors influencing the observed shieldings. In Chapter 1, current theories of nuclear magnetic shielding are briefly reviewed. A survey of various semi-empirical molecular orbital treatments of magnetic shielding is presented in Chapter 2 with particular emphasis on those employed in the present work, namely Pople's GIAO-MO procedure and the 'Uncoupled Hartree-Fock' method. In Chapters 3-6, Pople's GIAO-MO method has been successfully applied for the calculation of chemical shifts, shielding constants and their anisotropies for B, C, N, 0 and F nuclei in a variety of electronic environments. Reasonable agreement with experimental data is obtained in most cases. The results of comparable calculations based on the 'Uncoupled Hartree-Fock' method are also reported for comparison purposes. Finally, by means of selected examples, it has been demonstrated that a study of contributions to the paramagnetic shielding tensor due to variolas singlet electronic transitions may lead to a better understanding of the various factors governing the observed shieldings.