In this talk, I will discuss what is meant by "quantum symmetry" from the algebraic viewpoint of actions and coactions on algebras. The term "quantum" is used as algebras are allowed to be noncommutative here, say to include deformations of coordinate rings of varieties. I will mention some interesting results on when actions on algebras must factor or do not factor through actions of classical gadgets (such as groups or Lie algebras), that is, when we must enter the realm of quantum groups (or Hopf algebras) to understand symmetries of a given algebra. This all fits neatly into the framework of studying algebras in monoidal categories, and if time permits, I will give some recent results in this direction. I aim to keep the level of the talk down-to-earth by including many basic definitions and examples. Speaker(s): Chelsea Walton (University of Illinois)