The theory for the calculation of nuclear shielding constants with London atomic orbitals using density functional theory is presented. The theory includes the use of a local exchange-correlation functional which depends on both the electron density rho(r) and the paramagnetic current density j(p)(r). The resulting coupled-perturbed Kohn-Sham equations are presented, together with the working expression for the nuclear shielding tenser. The entire theory has been programmed and exhaustively checked, using standard Gaussian basis sets. A variety of rho(r) dependent exchange-correlation functionals have been used, together with a current dependence suggested by Vignale, Rasolt, and Geldart. Certain numerical difficulties arose with this form which necessitated a cutoff in its evaluation for low densities. Calculations have been performed on HF, N-2, CO, F-2, H2O, and CH4. Including the current dependence is seen here to have a slightly deshielding effect. The major deficiency in the reported calculations appears to arise from the use of local functionals which give poor values for the Kohn-Sham eigenvalues.