We present the results from a survey of 57 low-redshift Abell galaxy dusters to study the radial dependence of the luminosity function (LF). The dynamical radius of each cluster, r sub(200), was estimated from the photometric measurement of cluster richness, Bgc. The shape of the LFs is found to correlate with radius such that the faint-end slope, alpha , is generally steeper on the cluster outskirts. The sum of two Schechter functions provides a more adequate fit to the composite LFs than a single Schechter function. LFs based on the selection of red and blue galaxies are bimodal in appearance. The red LFs are generally flat for -22 less than or equal to M sub(RC) less than or equal to -18, with a radius-dependent steepening of alpha for M sub(RC) > -18. The blue LFs contain a larger contribution from faint galaxies than the red LFs. The blue LFs have a rising faint-end component ( alpha similar to -1.7) for M sub(RC) > -21, with a weaker dependence on radius than the red LFs. The dispersion of M* was determined to be 0.31 mag, which is comparable to the median measurement uncertainty of 0.38 mag. This suggests that the bright end of the LF is universal in shape at the 0.3 mag level. We find that M* is not correlated with cluster richness when using a common dynamical radius. Also, we find that M* is weakly correlated with BM type such that later BM-type clusters have a brighter M*. A correlation between M* and radius was found for the red and blue galaxies such that M* fades toward the cluster center.