A classical mathematical problem asks when two quadratic/bilinear/sesquilinear forms are equivalent by a change of coordinates (e.g., graphing a conic section). Residents of Hilbert space interpret this as a question about unitary orbits of operators, for which they offer beautiful answers in certain cases and a resigned shrug in general. The Weyl-von Neumann theorem is the first of many results showing that approximate unitary equivalence can be more accessible and similarly meaningful. By working instead with the norm closure of the unitary orbit, one removes measure theoretic information from an operator and can distill down, elegantly and powerfully, to the C*-algebraic information. There are variations in all directions: different topologies, domains, codomains, orbits. I will give a very selective tour of some history and recent results. A motivating question: what information is carried by the weak*-closed unitary orbit, and are good descriptions available? Among the novelties, I will exhibit 1) a convexity dichotomy and 2) a general correspondence between topologies and other kinds of "operator information". Some of this is joint work with Chuck Akemann.