  We outline a partial historical summary of the steps through which the
nucleosynthesis phenomena induced by {\it slow} neutron captures (the {\it
s-process}) were clarified, a scientific achievement in which Franz K\"appeler
played a major role. We start by recalling the early phenomenological approach,
which yielded a basic understanding of the subject even before models for the
parent stellar evolutionary stages were developed. Through such a tool, rough
limits for the neutron density and exposure were set, and the crucial fact was
understood that more than one nucleosynthesis component is required to account
for solar abundances of $s$-process nuclei up to the Pb-Bi region. We then
summarize the gradual understanding of the stellar processes actually involved
in the production of nuclei from Sr to Pb (the so-called {\it Main Component},
achieved in the last decade of the past century and occurring in red giants of
low and intermediate mass, ($M \lesssim$ 8 $M_{\odot}$), populating, in the
{\it HR} diagram, the {\it Asymptotic Giant Branch} or {\it AGB} region. We
conclude by giving some details on more recent research concerning mixing
mechanisms inducing the activation of the main neutron source,
$^{13}$C($\alpha$,n)$^{16}$O.
