The QCD axion remains one of the best-motivated proposals for the nature of dark matter. On the largest scales, axion dark matter is extremely cold and effectively pressureless, making it very similar to the WIMP and other candidates. However, as an extremely light scalar particle, axions can attain large occupation numbers, forming overdensities known as miniclusters, which can further collapse to form axion stars. In this story, the short-range self-interactions of the axion field are crucial. This “small scale structure” of axion dark matter can be an important distinguishing feature of the model. In this talk, I will review the recent advances and open questions in this field.