The difficulties of the phenomenological analysis of the weak s-compon
ent and the need for a comparison with the astrophysical picture are d
iscussed. New results of s-process calculations in massive stars durin
g both core He and shell C burning are presented. They are obtained af
ter adopting an updated nuclear network, which includes the deviation
from the 1/v trends of neutron capture cross sections, as well as an e
xtrapolation of beta decay rates to the high temperatures characterizi
ng carbon burning. For a complete discussion of the branching at Kr-85
, we additionally present a new measurement of the ''Rb neutron captur
e 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 mas
s range from Fe to Sr. Consequently, the results of s-processing in lo
w-mass stars during the thermally pulsing asymptotic giant branch phas
e are also considered, starting from the same set of revised nuclear p
arameters. 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 insi
de their present uncertainties, and by considering an additional contr
ibution to [sigma(n,gamma)](Kr-86] at low temperature by direct captur
es or undetected resonances. A detailed analysis of the neutron flow a
t branchings and at bottleneck points is performed, stressing the anal
ogies and differences between the phenomenological and astrophysical p
icture.