Let K be a knot in S^3 with hyperbolic complement. The seminal work of Thurston, and Culler-Shalen established the SL(2,C)-character variety X(K) as a powerful tool in the study of the topology of M. The canonical  component C is a component of X(K) that contains the character of a faithful discrete representation. Thurston proved that the canonical component is a curve. In this talk we will discuss some recent work in the direction of trying to understand what “these curves look like” as well as properties of these curves, their relation to the topology of S^3\K and arithmetic properties of Dehn surgeries on K.