There is an interesting interplay between continuum quantum field theory (QFT) and lattice systems. First, as in condensed-matter physics, we start at short distances (UV) with a lattice model and our goal is to find its long distance (IR) behavior. The lore is that this behavior is captured by a continuum QFT. Conversely, as is more common in high-energy physics and mathematical physics, the lattice theory is a first step toward a rigorous definition of the continuum theory. Despite enormous progress over the past decades, these two directions of the interplay between the lattice and the continuum face interesting challenges. Here, motivated by recently discovered theoretical phases of matter (including the XY-plaquette model and models of fractons), we will address two aspects of the relation between the lattice in the UV and the continuum in the IR. We will present lattice models exhibiting topological properties of continuum theories, like winding symmetries, ‘t Hooft anomalies, and duality. We will use this approach to clarify the subsystem global symmetries of some of the recently discovered exotic models. We will also discuss some more dynamical aspects of these systems and in particular their enigmatic UV/IR mixing; i.e., some long-distance properties are sensitive to short-distance details.