The word "Kondo effect'' was originally coined to indicate the behavior of magnetic alloys, but nowadays it is invoked, not always properly, Anytime a zero-bias anomaly is observed in the tunneling spectrum across a nanocontact. There is by now plenty of evidence of the conventional fully-screened Kondo effect in tunneling across quantum dots, single atoms and molecules. More recently, evidence of the so-called underscreened Kondo effect, an unusual phenomenon in magnetic alloys, has been also found in single-molecule transistors. In this talk, I will consider the possibility that nanocontacts could realize another phenomenon known to arise in magnetic alloys: the so-called "giant moment''. Indeed, while e.g. Fe diluted in Ag or Cu is known to undergo perfect Kondo screening, when diluted in Pd or Pt it seems not only to retain its magnetic moment but also to induce locally in the host ``giant moments'' that eventually order ferromagnetically at low temperature. In particular, I will briefly present what is meant by "giant moments'', and review theoretical explanations of the absence of Kondo screening in these alloys, checking their validity by a Numerical Renormalization group calculation. The ultimate goal is to identify the key properties of these alloys and to use this knowledge to anticipate the signals of "giant moments'' in transport across nanocontacts