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.
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