fsIn Part I, the theoretical basis for the calculation of electron spin resonance hyperfine coupling constants is outlined. The esr coupling constants are related to the wavefunction of a radical. The methods used in the calculation of the wavefunction are the Restricted Hartree Fock(RHF) method, the Unrestricted Hartree Fock (UHF) method, the UHF method with spin annihilation (i.e. the UHFAA method), the Configuration Interaction (CI) method, the approach of Da Providencia, and the Iterative Natural Orbital (INO) method. The Born-Oppenheimer approximation is used. In Part II, a comparison is made between the above methods using minimal basis sets of self consistent field atomic orbitals. The radicals used are CN, HCN-, HBO-, HCO, H, CN, BH3-, CH3, NH3+, NH2. Calculations on the radicals BOand BS were also performed. In Part III, the UHFAA method is used to perform calculations on the following radicals:- PH4, P2 H6+, PFn H4-n (n=0,1,2,3,4), PF5-, O23-, F2-, MgOH, ClF4, t-butyl radical in order to obtain information regarding the structural properties and/or identity of the above radicals.