A detailed examination is made of the surface abundance changes (particularly of the CNO nuclei) which are predicted and observed to occur in evolving globular cluster (GC) stars. Discussed first are the standard stellar evolutionary sequences which have been constructed for a mass of 0.8 an d a metallicity of Z = 1CT 4 ([Fe/H] = -2.25), for comparison with observations of extremely metal-deficient cluster stars, and for a mass of 0.9 3K 0 and a metal content of Z = 6 X 10 3 ([Fe/H] = -0.49), for comparison with data from relatively metal-rich systems. These calculations illustrate the well-known problem that canonical models are completely inadequate insofar as being able to explain either the observed steep decline of the carbon abundance as a function of luminosity along the giant branches of M15 and M 92 or the CN-bimodality which is seen among giant stars of similar magnitude and color in several intermediate-metallicity and metal-rich GCs (e.g., NCG 362, 47 Tue).Then, on the premise that enhanced mixing of the envelope into the stellar interior is responsible for the observed abundances, partially-mixed models have been constructed in which artificially deep mixing is initiated at a wide variety of evolutionary phases and maintained during the subsequent evolution. The purpose of these calculations was not only to attempt to reproduce theoretically the abundance data but also to try to determine when the mixing could occur without producing accompanying side effects that violated other observational constraints. The present work shows, for instance, that the onset of deep mixing, of a sufficient degree to alter the surface composition in the requisite way, at any time between the zero-age main sequence (ZAMS) and the turnoff will induce a significantly larger color spread on the C-M plane than is generally permitted by recent CCD photometry. Indeed, in the case of 47 Tue, it is argued that the tightness of its C-M diagram, the reported good agreement between its observed luminosity function with that derived from canonical models, and the inferred small spread in the star-to-star helium abundances among the cluster horizontal-branch stars all provide persuasive evidence that unusual mixing in postmain-sequence stars does not occur in this system. Worthy of note, however, is the suggestion from our analysis that a mixing explanation for the CN-bimodality in GCs may be least problematical if the abundance anomalies arise during the very early evolutionary history of a star. What would seem to be required is that stars which presently show strong CN must have remained fully mixed as they initially contracted toward the ZAMS and somewhat beyond, until n( 12 C) ~ n( 14 N), and then evolved normally. However, even if stars can remain fully mixed for the time required, this possibility is not without its difficulties. Consequently, it is highly probable that some explanation other than mixing is responsible for the abundance anomalies which are seen in the bimodal-CN clusters. In contrast, the observed tr...