A classical problem in elimination theory is to find the implicit equations defining the graphs and images of rational maps between projective varieties. The bi-homogeneous coordinate ring of the graph of such maps is the Rees ring of the ideal $I$, generated by the forms that define the map. The homogeneous coordinate ring of the image—the variety parametrized by the map—is the special fiber ring of the same ideal $I$. Thus, the goal becomes to determine the defining ideal of the Rees ring, and thereby of the special fiber ring. This question has been addressed in well over a hundred articles by commutative algebraists, algebraic geometers, and applied mathematicians. The problem is difficult, and each class of ideals (or rational maps) seems to require different techniques. We will survey some of the results in this area focusing mainly on the different methods.