  The cosmic ray energy spectra encode very important information about the
mechanisms that generate relativistic particles in the Milky Way, and about the
properties of the Galaxy that control their propagation. Relativistic electrons
and positrons traveling in interstellar space lose energy much more rapidly
than more massive particles such as protons and nuclei, with a rate that grows
quadratically with the particle energy $E$. One therefore expects that the
effects of energy loss should leave observable signatures in the $e^\mp$
spectra, in the form of softenings centered at the critical energy $E^*$. This
quantity is determined by the condition that the total energy loss suffered by
particles during their residence time in the Galaxy is of the same order of the
initial energy. If the electrons and positrons are accelerated in discrete
(quasi) point--like astrophysical objects, such as Supernova explosions or
Pulsars, the stochastic nature of the sources should also leave observable
signatures in the $e^\mp$ energy spectra and angular distributions above a
second (higher) critical energy $E^\dagger$, determined by the condition that
particles with $E \gtrsim E^\dagger$ can propagate only for a maximum distance
shorter than the average separation between sources. In this work we discuss
the theoretically expectations for the signatures of the energy loss effects on
the electron and positron spectra, and compare these predictions with the
existing observations. Recent measurements of the $(e^- + e^+$) flux have
discovered the existence of a prominent spectral break at $E \simeq 1$~TeV.
This spectral feature can perhaps be identified with the critical energy $E^*$.
An alternative hypothesis is to assmume that $E^*$ has a much lower value of
order few GeV. Resolving this ambiguity is of great importance for our
understanding of Galactic cosmic rays.
