From a directed graph Q, called a quiver, one can construct what is known as a quiver variety Y_Q, an algebraic variety defined as a quotient of a vector space by a group defined in terms of Q.  A mutation of a quiver is an operation that produces from Q and new directed graph Q’ and a new associated quiver variety Y_{Q’}.  The mutation conjecture predicts a surprising and beautiful connection between the geometry of Y_Q and that of Y_{Q’}.  In this talk I will describe quiver varieties and mutations, and show you that you are already well acquainted with some examples of these.  Then I will discuss an interesting connection to the Gromov—Witten Theory of degeneracy loci. This is based on joint work with Nathan Priddis and Yaoxiong Wen.