  Recent precision measurements of the flux of cosmic ray positrons by the
Alpha Magnetic Spectrometer show that the spectrum has a marked softening
feature for energies close to one TeV. A possible interpretation of this result
is that the observed feature measures the maximum energy of a new hard source
of positrons perhaps associated to dark matter self--annihilation or decay, or
to positron accelerators. A gradual hardening of the positron flux centered at
$E \simeq 25$~GeV can also be understood as the signature of the transition
where the new source overtakes the conventional component due to secondary
production. This interpretation is simple and attractive, but it is not unique.
The alternative possibility, that the positron flux is entirely of secondary
origin, remains viable. In such a scenario the spectral softening observed by
AMS for positrons is generated by energy loss effects, and a feature of
similar, but not identical structure should be also visible in the $e^-$
spectrum. Spectral features similar to both the hardening and softening of the
positron flux are in fact observed for electrons and call for a consistent
explanation. Precision measurements of the $e^+$ and $e^-$ spectra in the TeV
and multi--TeV energy range are crucial to clarify the problem.

High Energy Astrophysical Phenomena