Exchange polarization of core electrons by outer unpaired electrons has been calculated for 10 different atomic configurations of Li, Na, K, F, Cl, Be, B, and N in the unrestricted Hartree-Fock (UHF) approximation. Numerical integration techniques were used and accurate conventional Hartree-Fock (HF) wave functions were also obtained for these configurations. The theory of atomic hyperfine structure in the UHF approximation is developed and the HF and UHF calculated values of the hyperfine coupling constants are compared with available experimental data. The importance of core polarization in solid state problems is briefly mentioned with particular attention to color centers. Finally, unsuccessful attempts to calculate core polarization by perturbation expansion methods are discussed.