We design algorithms for finding roots of polynomials over function fields of curves. Such algorithms are useful for list decoding of Reed-Solomon and algebraic-geometric codes. In the first half of the paper we will focus on bivariate polynomials, i.e., polynomials over the coordinate ring of the affine line. In the second half we will design algorithms for computing roots of polynomials over the function field of a nonsingular absolutely irreducible plane algebraic curve. Several examples are included