  This is a review, intended for lattice nonspecialists, of the studies of the
compact abelian gauge theories on the lattice performed by the Aachen lattice
field theory group. We discuss in particular the pure compact QED and a U(1)
lattice gauge theory with charged scalar and fermion fields in four and three
dimensions. Our data indicate that these lattice theories might define some
continuum field theories, or at least effective field theories with remarkable
nonperturbative properties like confinement and dynamical mass generation.
