  In recent years, space-born experiments have delivered new measurements of
high energy cosmic-ray (CR) $\bar p$ and $e^+$. In addition, unprecedented
sensitivity to CR composite anti-nuclei anti-d and anti-He is expected to be
achieved in the near future. We report on the theoretical interpretation of
these measurements. While CR antimatter is a promising discovery tool for new
physics or exotic astrophysical phenomena, an irreducible background arises
from secondary production by primary CR collisions with interstellar matter.
Understanding this irreducible background or constraining it from first
principles is an interesting challenge. We review the attempt to obtain such
understanding and apply it to CR $\bar p,\, e^+,$ anti-d and anti-He. Based on
state of the art Galactic cosmic ray measurements, dominated currently by the
AMS-02 experiment, we show that: (i) CR $\bar p$ most likely come from CR-gas
collisions; (ii) $e^+$ data is consistent with, and suggestive of the same
secondary astrophysical production mechanism responsible for $\bar p$ and
dominated by proton-proton collisions. In addition, based on recent accelerator
analyses we show that the flux of secondary high energy anti-He may be
observable with a few years exposure of AMS-02. We highlight key open
questions, as well as the role played by recent and upcoming space and
accelerator data in clarifying the origins of CR antimatter.
