We classify elliptic surfaces of Kodaira dimension zero by completing the
Bombieri–Mumford list of possible configurations of invariants, which was obtained
by means of the canonical bundle formula and other restrictions. In the characteristic
zero case, an example of each configuration can be constructed by means of
logarithmic transformation and algebraization. In the positive characteristic case, we
show that some configurations do not appear, and we determine which configurations
actually appear.
