Evolutionary tracks have been obtained for Population II red giants for the following sets of (M, X, Z): (0.80 Mo., 0.7, l0- , (0.80 M5, 0.7, l0- ), (0.80 Al0, 0.8, l0- ), (0.60 M0, 0.7, l0- ), (0.80 M0, 0.7, 10-2). The mass of the helium core at the flash is found to be M1M0 = 0.475 + 0.23 (X - 0.7) - 0.01 (log Z + 3) - 0.035 (MIM0 - 0.8). If direct electron-neutrino interactions are neglected, the core masses are 0.033 Ma, larger. The lifetimes on the giant branch above a given luminosity are also determined. These are somewhat lower than in earlier work. Small bumps are predicted in red-giant luminosity functions.