Symmetry is powerful principle in physics, allowing us to make exact statements even in regimes where controlled calculations are challenging or impossible.  Thus understanding the ways in which different types of symmetries can constrain phases of matter is an important component of understanding what nature is capable of.  In this talk, I will describe how exploring new types of symmetries, including symmetries with unusual spatial structure, or symmetries that act on particles in exotic ways, has expanded our understanding of these possibilities, including identifying new classes of phases of matter, and new platforms with which to realize these.