For a space X, the n-th topological complexity (TC) of X, was defined by Rudiak and is related to robotic motion planning. Farber and Oprea put these into a generating function, and conjectured that it has a single pole of degree 2 at x=1. After introducing the basic notions of TC, we show that this conjecture holds for some class of spaces. We also related this to counterexamples to Ganea's conjecture first constructed by Iwase. We describe joint work with Michael Farber and Daisuke Kishimoto and also with Jeff Strom. All of this work was done during the Fields program.