  Fracton theories possess exponentially degenerate ground states, excitations
with restricted mobility, and nontopological higher-form symmetries. This paper
shows that such theories can be defined on arbitrary spatial lattices in three
dimensions. The key element of this construction is a generalization of
higher-form gauge theories to so-called $\mathfrak{F}_p$ gauge theories, in
which gauge transformations of rank-$k$ fields are specified by rank-$(k - p)$
gauge parameters. The $\mathbb{Z}_2$ rank-two theory of type $\mathfrak{F}_2$,
placed on a cubic lattice and coupled to scalar matter, is shown to have a
topological phase exactly dual to the well-known X-cube model. Generalizations
of this example yield novel fracton theories. In the continuum, the
$\mathrm{U}(1)$ rank-two theory of type $\mathfrak{F}_2$ is shown to have a
perturbatively gapless fracton regime that cannot be consistently interpreted
as a tensor gauge theory of any kind. The compact scalar fields that naturally
couple to this $\mathfrak{F}_2$ theory also show gapless fracton behavior; on a
cubic lattice they have a conserved $\mathrm{U}(1)$ charge and dipole moment,
but these particular charges are not necessarily conserved on more general
lattices. The construction straightforwardly generalizes to $\mathfrak{F}_2$
theories of nonabelian rank-two gauge fields, giving first examples of pure
nonabelian higher-rank theories.
