Independence polynomials are generating functions for the number of independent sets of each cardinality in a graph G. In addition to encoding useful information about the graph (such as the number of vertices, the number of edges and the independence number), the analytic and algebraic properties can say much about the shape and inter-dependence of the coefficients. In this talk well focus on the nature and location of the roots of such polynomials, and even cross paths with a fractal or two! This research is joint with Ben Cameron, Iain Beaton, Karl Dilcher, Richard Hoshino and Richard Nowakowski.