The difficulties of the phenomenological analysis of the weak s-component and the need for a comparison with the astrophysical picture are discussed. New results of s-process calculations in massive stars during both core He and shell C burning are presented. They are obtained after adopting an updated nuclear network, which includes the deviation from the 1/v trends of neutron capture cross sections, as well as an extrapolation of beta decay rates to the high temperatures characterizing carbon burning. For a complete discussion of the branching at Kr-85, we additionally present a new measurement of the ''Rb neutron capture cross section. The reproduction of the weak component by successive stellar generations is investigated through a global check which takes into account the interference of the main component in the atomic mass range from Fe to Sr. Consequently, the results of s-processing in low-mass stars during the thermally pulsing asymptotic giant branch phase are also considered, starting from the same set of revised nuclear parameters. A satisfactory match of the solar s-abundances is reached, where the problem of the Kr-86 overproduction by the main component is overcome by varying the [sigma(n, gamma)] of Kr-84, Kr-85, Kr-86 inside their present uncertainties, and by considering an additional contribution to [sigma(n,gamma)](Kr-86] at low temperature by direct captures or undetected resonances. A detailed analysis of the neutron flow at branchings and at bottleneck points is performed, stressing the analogies and differences between the phenomenological and astrophysical picture.