I will briefly describe the Cox ring of a projective variety, which is an important feature of its birational geometry. Afterwards, I sketch the a counterexample for Hilbert's 14th problem constructed by Shigeru Mukai in 2003 as an application of Cox rings, by showing that under some conditions the Cox ring of a blowup of projective space at some number of points will not have a finitely generated Cox ring. Speaker(s): Saket Shah (UM)