  We show that most classes of K3 surfaces have only finitely many Enriques
quotients. For supersingular K3 surfaces over fields of characteristic $p \geq
3$, we give a formula which generically yields the number of their Enriques
quotients. We reprove via a lattice theoretic argument that supersingular K3
surfaces always have an Enriques quotient over fields of small characteristic.
For some small characteristics and some Artin invariants, we explicitly compute
lower bounds for the number of Enriques quotients of a supersingular K3
surface. We show that the supersingular K3 surface of Artin invariant $1$ over
an algebraically closed field of characteristic $3$ has exactly two Enriques
quotients.
