The asymptotic giant branch evolution of stars prior to the onset of thermal pulses is found to be highly sensitive to both initial composition and total stellar mass. Models investigated have masses in the range 3--11 M/sub sun/ and initial compositions of (Y,Z) = (0.28, 0.001), (0.28, 0.01), (0.28, 0.02), (0.28, 0.03), (0.20, 0.02), (0.36, 0.02), and (0.20, 0.001). The effects produced by varying the overall rate of neutrino emission and by varying the size of the mixing length in convective regions are studied; the role played by convective overshoot, semiconvection, and the energy cost of mixing across the hydrogen-helium discontinuity are discussed. a general description of the evolutionary behavior of our models from the start of the asymptotic giant branch to the first major thermal pulse is also presented.The inward advance of the base of the convective envelope results in a reduction in the size of the hydrogen-exhausted core for stellar masses in a specific range: M/sub UP/> or =M/sub asterisk/> or =M/sub CRIT/. The magnitudes of both M/sub CRIT/ and M/sub UP/ (defined as the minimum mass for nondegenerate carbon ignition) depend on the initial composition in ways that can be approximated analytically. The final size of themore » hydrogen-exhausted core that results after the dredge-up phase is, for a given initial mass, larger in models with a lower A or higher Y. The abundance changes that occur as a result of dredge-up are, for a given mass, greater in models with a lower Z or higher Y. Typically, /sup 4/He and /sup 14/N are enhanced while /sup 1/H, /sup 12/C, and /sup 16/O are depleted. For the most massive models undergoing dredge-up, significant enhancements in /sup 18/O also occur.« less