Abundances of some 20 elements have been determined for a (biased) sample of 40 red giants having M(v) < -1.5 in the chemically inhomogeneous globular cluster omega Centauri. The results are based on high-resolution, high signal-to-noise echelle spectra and permit one to examine the roles of primordial enrichment and stellar evolutionary mixing effects in the cluster. Our basic conclusions are as follows (1) There is an abundance range -1.8 < [Fe/H] < -0.8, and even more metal rich stars may exist in the cluster. (2) For the alpha (Mg, Si, Ca, Ti) and iron peak (Cr, Ni) elements and Sc and V, [metal/Fe] is flat as a function of [Fe/H] and is consistent with primordial enrichment from stars having mass greater than 10 M., as has been found for field halo stars. (3) There is a large scatter in the abundances of C, N, and O. The bulk of the stars have -0.9 < [C/Fe] < -0.3 and [O/Fe] similar to 0.3, as is found at the red giant branch tip in other ''normal'' (showing no spread in [Fe/H]) clusters of similar abundance, while there also exists a group of CN-strong stars having [C/Fe] similar to -0.7 and [O/Fe] similar to -0.5. Nitrogen appears to be enhanced in all of these carbon-depleted stars. These results are most readily explained in terms of evolutionary mixing effects not predicted by standard stellar evolution calculations and are consistent with the earlier suggestions of Cohen & Bell (1986) and Paltoglou & Norris (1989) concerning processing in both the CN and ON cycles in the stars being observed. In contrast, the group of CO-strong stars first identified by Persson et al. (1980) has [C/Fe] similar to 0.0, [O/Fe] - 0.4, and [N/Fe] similar to 0.4 (or 0.9 if the nitrogen scale of Brown and Wallerstein is correct) and is suggestive of primordial enrichment of carbon and/or nitrogen from intermediate- and possibly low-mass stars, tempered by later stellar evolutionary effects. (4) [Na/Fe] and [Al/Fe] are anticorrelated with [O/Fe], and there is a positive correlation between [Na/Fe] and [Al/Fe], all of which are most readily explained in terms of evolutionary mixing effects as first suggested by Denisenkov and Denisenkova (1990). Such an explanation is supported by the similar ([Na/Fe], [O/Fe]) anticorrelation reported by Kraft et al. (1993) in the ''normal'' globular clusters. (5) For the heavy neutron-addition elements (in particular Y, Ba, La, and Nd) [heavy metal/Fe] rises as [Fe/H] increases, in sharp contrast with what is found in the ''normal'' clusters, while the relative abundances as a function of atomic number are suggestive of s-processing. The increase in [heavy metal/Fe] with [Fe/H] appears independent of the abundance of C, N, O, Na and Al and is most naturally explained as a primordial effect. Guided by the predictions of existing (somewhat uncertain) stellar evolution calculations, we suggest that this results from primordial enrichment from stars having mass as low as 1-3 M.. (6) If the preceding suggestion is correct, chemical enrichment in omega Cen occurred over an extended period, perhaps greater than or equal to 1 Gyr.