Spacetime and global symmetries play a fundamental role in modern physics. Among other things, they lead to selection rules and non-renormalization theorems which hold for any theory with that symmetry. Remarkably, the spacetime symmetries of Galilean-invariant, non-relativistic theories have only been understood in the past year. I will discuss this progress as well as some implications for transport phenomena and a new spin on Schrodinger-invariant field theories.