An important aspect of quantum mechanics is that it introduces non-commutative phenomena into physics: Heisenberg's uncertainty principle reflects the fact of mathematical life that matrices don't necessarily commute. Over the last 80 years the study of non-commutative structures has become an increasingly popular activity in many branches of mathematics. The two examples I will talk about are non-commutative topology and non-commutative measure theory. My main goal will be to explain what people mean when they use these words and why the terminology is justified. I hope this will be interesting for geometers and analysts alike. The main ingredients in non-commutative geometry are non-commutative rings, so if you're an algebraist, maybe you'll find something to like as well. Time permitting I'll describe some examples and theorems relating to the classification of non-commutative measure spaces (a.k.a. von Neumann algebras).