We present a grid of theoretical evolutionary models covering the horizontal-branch (HB) and the asymptotic giant branch (AGB) phases of globular cluster stars. The computations have been performed for a fixed amount of the original He (Y = 0.23) and for the metallicity values Z = 10(-4), 4 x 10(-4), 10(-3), 6 x 10(-3) and 2 x 10(-2). For each value of the assumed metallicity, selected zero-age horizontal-branch models with evolutionary values for both M(c) (mass of the He core) and dY (extra helium at the surface) have been evolved from the initial He-burning structure trough the whole phase of central He burning and along the following He shell burning phase until the minimum in luminosity marking the reignition of the H shell and the onset of the thermally pulsating phase. We provide extensive tabulations of the numerical results to be compared with globular cluster observations. We find that the luminosity of the AGB clump and the number ratio of AGB to HB stars should remain fairly constant with the cluster metallicity. However, the difference in bolometric luminosity between the AGB clump and the HB should be a rather sensitive function of the cluster metallicity, increasing when the cluster metallicity increases. We discuss the dependence of the ZAHB luminosity at log T(e) = 3.85 on the assumed metallicity. We find that at the lower metallicies the slope of this luminosity versus Z is rather similar to the value given in previous theoretical evaluations. However, for Z larger than 0.001 this slope becomes larger, approaching the value suggested by Sandage on observational grounds. As a result, we find that for a given cluster age the difference in bolometric magnitudes between the HB and the turnoff should be rather insensitive to the cluster metallicity. We finally present a recalibration of the parameter R, suggesting a small increase in the adopted value of the original He in Galactic globulars and, perhaps, giving a suggestion for a possible correlation of helium and metals in 47 Tuc.
