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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.