  We classify globally irreducible representations of alternating groups and
double covers of symmetric and alternating groups. In order to achieve this
classification we also completely characterise irreducible representations of
such groups which reduce almost homogeneously in every characteristic. This
also allows us to classify irreducible representations that remain irreducible
in every characteristic. In particular we show that, apart from finitely many
exceptions, for any of these three questions such representations are either
$1$-dimensional or basic spin representations.
