  Motivated by the prospects for an improved test of lepton universality in K
-> l \nu decays by the NA62 experiment at CERN, we study predictions for the
possible lepton non-universality in K -> l \nu decays in supersymmetric models.
Violations of \mu-e universality in this process may originate from mixing
effects in the right-handed slepton sector, providing a unique window into this
aspect of supersymmetric flavour physics in the large-\tan\beta region. Minimal
unification scenarios with universal soft supersymmetry-breaking terms at the
GUT scale would predict negligible violation of lepton universality. However,
lepton non-universality may be observable in non-minimal grand unified models
with higher-dimensional terms contributing to fermion masses, in which case
renormalization effects above the GUT scale may enhance the mixing among the
right-handed sleptons. This could leads to observable lepton non-universality
in K -> l \nu decays in specific regions of the parameter space with high \tan
\beta, large A terms and small charged Higgs boson mass. Observable
non-universality in K -> l \nu decays would be correlated with a large value of
BR(\tau -> e \gamma). The experimental upper limit on the electric dipole
moment of the electron could be reconciled with leptogenesis, if the latter
occurs at a relatively low scale, which would also alleviate the cosmological
gravitino problem. Even if lepton non-universality is not seen in the near
future, one may nevertheless obtain significant constraints on the model
parameters and unknown aspects of right-handed fermion and sfermion mixing.
