  We reformulate known exotic theories (including theories of fractons) on a
Euclidean spacetime lattice. We write them using the Villain approach and then
we modify them to a convenient range of parameters. The new lattice models are
closer to the continuum limit than the original lattice versions. In
particular, they exhibit many of the recently found properties of the continuum
theories including emergent global symmetries and surprising dualities. Also,
these new models provide a clear and rigorous formulation to the continuum
models and their singularities. In appendices, we use this approach to review
well-studied lattice models and their continuum limits. These include the
XY-model, the $\mathbb{Z}_N$ clock-model, and various gauge theories in diverse
dimensions. This presentation clarifies the relation between the
condensed-matter and the high-energy views of these systems. It emphasizes the
role of symmetries associated with the topology of field space, duality, and
various anomalies.
