Oxygen isotopic measurements of 20 spinel (MgAl(2)O(4)) grains from the CM2 meteorite Murray (average diameter 0.45 mu m), seven spinel grains from ordinary chondrites (OC, 0.3 to 2 Am) and three spinel grains from the CI chondrite Orgueil (0.4 to 0.7 mu m) have revealed large anomalies and thus established their presolar origin. Their O isotopic ratios fall into all four previously defined groups and indicate that most of the grains come from red giant branch (RGB) or asymptotic giant branch (AGB) stars. With the NanoSIMS, we measured the magnesium isotopic compositions of all 30 grains. We also measured the Cr isotopic ratios of the three Orgueil grains and one OC grain. Two of the Orgueil grains are very rich in Cr, with a composition representing a 1:1 solid solution of spinel and magnesiochromite (MgCr(2)O(4)). At least 13 of the 30 analyzed grains have substantial Mg isotopic anomalies. They have excesses in (26)Mg, primarily from the decay of (26)Al, as well as excesses and deficits in (25)Mg. We present the results of new model calculations of the evolution of Mg and Al isotopic ratios in the envelopes of AGB stars for a range of masses (1.5, 2, 3, and 5 M(circle dot)), metallicities (1/6, 1/3, 1/2 and 1Z(circle dot)) and different prescriptions for mass loss by stellar winds. Comparisons of the grain data with these models show that inferred (26)Al/(27)Al ratios of several grains are much larger than predicted and require an extra production process for (26)Al, most likely the result of deep mixing in the star s envelope, also called cool bottom processing (CBP). One grain is probably from an intermediate-mass (similar to 5 M(circle dot)) AGB star in which the envelope extended down to the H-burning shell ( hot bottom burning ). On average, (26)Al/(27)Al ratios in oxide grains are larger than those in SiC grains from AGB stars (mainstream, Y and Z grains), whose (26)Al/(27)Al ratios agree with AGB models. This indicates that the parent stars of oxide grains with high (> 5 x 10(-3))(26)Al/(27)Al ratios are fundamentally different from those of SiC grains. The CBP experienced by the former might have prevented these stars from becoming the carbon stars that would have produced presolar SiC grains.