Mock theta function are a mysterious family of functions defined by Srinivasa Ramanujan. Ramanujan defined them in a letter to G. H. Hardy. Only part of his letter survives, so the actual definition of what makes a function a mock theta function has been lost. The surviving remnant contains examples which make the resemblace to theta functions clear.
I ran across this press release put out by the University of Wisconsin-Madison about a breakthrough in mock theta functions, and I had no idea what they were actually announcing. The best I can piece together is that Sander Zwegers, in his dissertation gave the first general definition of mock theta functions that included most of Ramanujan’s examples, and that this work was extended by Kathrin Bringmann and Ken Ono. Some details can be found in this preprint.