Ultraluminous X-ray sources (ULXs) are amongst the most intriguing of X-ray source classes. Their extreme luminosities - greater than 10^39 erg/s in the 0.3 - 10 keV band alone - suggest either the presence of black holes larger than those regularly encountered in our own Galaxy (the Galactic centre excepted), or sources apparently radiating well above the Eddington limit. We review the insights afforded us by studies of their X-ray emission, focussing on what this reveals about the underlying compact object. In particular, we discuss recent deep observations of ULXs by the XMM-Newton observatory, and how the unprecedented data quality provided by this mission is starting to discriminate between the different physical models for these extraordinary X-ray emitters. ; Comment: Invited talk at the 5th Stromlo Symposium. 10 pages, 4 figures. Accepted for publication in Astrophysics & Space Science