The laws of quantum mechanics have been known for nearly a century, and have passed every test that has been made of them. But knowing them does not mean that we understand all that they permit. Einstein noticed that a key prediction of quantum mechanics is that it permits states with what is now called long-range “entanglement;” in his “second biggest mistake,” he thought that this would not be seen experimentally, thus falsifying quantum mechanics. In recent years, it has gradually been realized that many strange classes of condensed matter can occur, which have robust unexpected exotic properties. Their “quantum entanglement” has turned out to be a key feature, suggesting applications to “quantum computing.” Their quantum states can be classified as being “topologically non-trivial,” distinct from “topologically-trivial” matter that can in principle be continuously evolved from widely-separated atoms. I will describe some of the simplest examples that led to this new understanding.