  The simplest explanation for the atmospheric neutrino anomaly is the
transition of muon neutrinos and antineutrinos into states of different flavor.
Data from the reactor experiment Chooz, and the measurement of the energy and
angle distribution of electron--like events in Super--Kamiokande put strong
constraints on $\nu_\mu$--$\nu_e$ mixing, but the transitions $\nu_\mu \to
\nu_\tau$ and $\nu_\mu \to \nu_s$ are viable possibilites. If the muon
neutrinos are mixed with a singlet sterile state, the energy and zenith angle
dependence of the oscillation probability is modified by matter effects, that
are important for $E_\nu/|\Delta m^2| \aprge 10^3$ GeV/eV$^2$. We confront the
theoretical predictions with existing data on contained events and
neutrino--induced upward going muons.
