I'll discuss two problems, which on the surface seem quite different. The first, which comes up in signal processing and in algorithm design, is the problem of coming up with linear, geometry-preserving maps which are efficient to store and manipulate. The second, which comes up in coding theory and theoretical computer science, is the problem of establishing the list-decodability -- a combinatorial property -- of error correcting codes. I'll establish a connection between these two problems, and discuss how techniques from high-dimensional probability can be used to handle both. Punchlines include improved fast Johnson-Lindenstrauss transforms and structured RIP matrices, and the answer to some longstanding combinatorial open questions in coding theory.