  This paper has a twofold purpose: to present an overview of the theory of
absolutely summing operators and its different generalizations for the
multilinear setting, and to sketch the beginning of a research project related
to an objective search of \textquotedblleft perfect\textquotedblright \
multilinear extensions of the ideal of absolutely summing operators. The final
section contains some open problems that may indicate lines for future
investigation.
