The Moon appears to have an iron-rich core, but we're not sure exactly how big. There is very limited seismic data from the Apollo missions in part because the Moon isn't very active and partly because we didn't crash nearly enough Saturn V third stages into the Moon. So not many earthquake waves have ever been recorded passing through the deep lunar interior - which appears to be very weird and sort of slushy deep down (stop me if I get too technical).
In 2010, a paper* reprocessed the Apollo data and suggest the Moon has a solid core with a radius of 330km +/- 20km. The mass of this core is uncertain because its composition is also unclear, but the usual iron-nickel alloy seems likely with up to 6% dissolved sulfur by weight.
* Weber, R. C.; Lin, P.-Y.; Garnero, E. J.; Williams, Q.; Lognonne, P. (2011). "Seismic Detection of the Lunar Core" (PDF). Science. 331 (6015): 309–312.