In this talk, I will discuss how microscopic driving and conservation laws shape the large-scale statistical physics of driven particle systems.

I will first focus on phase-separated active fluids, whose interfaces display a surprisingly rich phenomenology. I will show, numerically and theoretically, how the universality classes of these interfaces are affected by conservation laws and symmetries, including whether the underlying dynamics is chiral or achiral.

When the center of mass is conserved, the system as a whole, and not just its interface dynamics, exhibits unusual physical behavior rooted in an underlying hyperuniformity. I will show how this hyperuniformity reshapes the statistical mechanics of fluids and crystals, affecting also nucleation, phase separation, critical phenomena, or the Mermin-Wagner theorem.