  Recent numerical studies of the 4D pure compact U(1) lattice gauge theory, I
have participated in, are reviewed. We look for a possibility to construct an
interesting nonperturbatively renormalizable continuum theory at the phase
transition between the confinement and Coulomb phases. First I describe the
numerical evidence, obtained from calculation of bulk observables on spherical
lattices, that the theory has a non-Gaussian fixed point. Further the
gauge-ball spectrum in the confinement phase is presented and its universality
confirmed. The unexpected result is that, in addition to massive states, the
theory contains a very light, possibly massless scalar gauge ball. I also
summarize results of studies of the compact U(1) lattice theory with fermion
and scalar matter fields and point out that at strong coupling it represents a
model of dynamical fermion mass generation.
