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In this talk, I will discuss “quantum symmetry” from an algebraic viewpoint, especially for symmetries of algebras. The term “quantum” is used as algebras here are usually noncommutative. I will mention some interesting results on when symmetries of algebras must factor or do not factor through symmetries 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.