Operator algebras come in two types: von Neumann algebras and C*-algebras. Although they look similar at first sight, von Neumann algebras have a measure-theoretic flavor, while C*-algebras are more topological. In 1970s Alain Connes made stunning breakthroughs in understanding the structure and symmetries of simple injective von Neumann algebras. In the last couple of decades, an army of mathematicians have been working on C*-analogues of Connes' results, and we are now witnessing remarkable advances. I will discuss some of the highlights, including the recent development of non-commutative Rokhlin type theorems. This talk will be accessible to first-year graduate students. No prior knowledge of operator algebras is needed.