Over fields of characteristic other than 2, Artin (1960) showed that the theory is similar to that over the complex numbers.
Over fields of characteristic 2 the definition is modified, and there are two new families, called singular and supersingular Enriques surfaces, described by Bombieri & Mumford (1976).
These two extra families are related to the two non-discrete algebraic group schemes of order 2 in characteristic 2.
Hodge diamond: Marked Enriques surfaces form a connected 10-dimensional family, which Kondo (1994) showed is rational.
In characteristic 2 the definition of Enriques surfaces is modified: they are defined to be minimal surfaces whose canonical class K is numerically equivalent to 0 and whose second Betti number is 10.