We present a comprehensive theoretical investigation of the evolutionary properties of intermediate-mass stars. The evolutionary sequences were computed from the zero-age main sequence up to the central He exhaustion and often up to the phases that precede the carbon ignition or to the reignition of the H-shell, which marks the beginning of the thermal pulse phase. The evolutionary tracks were constructed by adopting a wide range of stellar masses (3 less than or equal to M/M. less than or equal to 15) and chemical compositions. In order to account for current uncertainties on the He to heavy elements enrichment ratio (DeltaY/DeltaZ), the stellar models were computed by adopting at Z = 0.02 two different He contents (Y = 0.27, 0.289) and at Z = 0.04 three different He contents (Y = 0.29, 0.34, and 0.37). Moreover, to supply a homogeneous evolutionary scenario that accounts for young Magellanic stellar systems the calculations were also extended toward lower metallicities (Z = 0.004, Z = 0.01), by adopting different initial He abundances. We evaluated for both solar (Z = 0.02) and super-metal-rich (SMR, Z = 0.04) models the transition mass M-up between the stellar structures igniting carbon and those that develop a full electron degeneracy inside the carbon-oxygen core. We found that M-up is Of the order of 7.7 +/- 0.5 M. for solar composition, while for SMR structures an increase in the He content causes a decrease in M-up, and indeed it changes from 9.5 +/- 0.5 M, at (Y = 0.29, to 8.7 +/- 0.2 M. at Y = 0.34, and to 7.7 +/- 0.2 M. at Y = 0.37. We also show that M-up presents a nonlinear behavior with metallicity, and indeed it decreases when moving from Z = 0.04 to Z approximate to 0.001 and increases at lower metal contents. This finding confirms the predictions by Cassisi & Castellani and more recently by Umeda et al. and suggests that the rate of SNe type Ia depends on the chemical composition of the parent stellar population. This approach allows us to investigate in detail the evolutionary properties of classical Cepheids. In particular, we find that the range of stellar masses that perform the blue loop during the central He-burning phase narrows when moving toward metal-rich and SMR structures. This evidence and the substantial decrease in the evolutionary time spent by these structures inside the instability strip bring out that the probability of detecting long-period Cepheids in SMR stellar systems is substantially smaller than in more metal-poor systems. Moreover, and even more important, we find that the time spent by Cepheids along the subsequent crossings of the instability strip also depends on the stellar mass. In fact, our models suggest that low-mass, metal-poor Cepheids spend a substantial portion of their lifetime along the blueward excursion of the blue loop, while at higher masses (M/M. greater than or equal to 8) the time spent along the redward excursion becomes longer. Models at solar chemical composition present an opposite behavior, i.e., the time spent along the redward excursion is longer than the blueward excursion among low-mass Cepheids and vice versa for high-mass Cepheids. Oddly enough, the time spent along the blueward excursion is for models at Z = 0.01 longer than the redward excursion over the entire mass range. This suggests a nonlinear dependence of crossing times on metallicity. The time spent along the first crossing of the instability strip is generally negligible with the exception of high-mass, metal-poor stellar structures for which it becomes of the order of 15%-20% of the total crossing time.
