  We extend our exploration of nonstandard continuum quantum field theories in
2+1 dimensions to 3+1 dimensions. These theories exhibit exotic global
symmetries, a peculiar spectrum of charged states, unusual gauge symmetries,
and surprising dualities. Many of the systems we study have a known lattice
construction. In particular, one of them is a known gapless fracton model. The
novelty here is in their continuum field theory description. In this paper, we
focus on models with a global $U(1)$ symmetry and in a followup paper we will
study models with a global $\mathbb{Z}_N$ symmetry.
