We extend the idea of Grunsky operator and Faber operator to higher genus compact Riemann surfaces with one boundary curve. In this talk, I will first review the classical concept of Faber Polynomials corresponding to the map $f$, a conformal map on a simply connected domain $\textbf{D}$ of the Riemann sphere with $\Gamma=\partial f (\textbf{D})$. We define the Grunsky coefficients associated to $f$, and review the necessary and sufficient conditions for $f$ to be univalent on $\textbf{D}$. We will define Grunsky and Faber operators and show that the Faber operator is an isomorphism for quasi-circles. I will present some recent work of myself, E. Schippers and W. Staubach generalizing these results to conformal maps into compact Riemann surfaces. Finally, a model which reveals some analogies between the Grunsky operator and the classical period mapping on the Universal Teichmuller space will be discussed.