Given an error-correcting code of block length n and an arbitrary input string also of length n, the list decoding problem is that of finding all codewords within a specified Hamming distance from the input string. We present an improved list decoding algorithm for decoding Reed-Solomon codes. The list decoding problem for Reed-Solomon codes reduces to the following ``curve-fitting'' problem over a field F: Given n points (x[i].y[i]), 1 <= i <= n (where x[i],y[i] are elements of the field F), a degree parameter k and error parameter e, find all univariate polynomials P over the field F of degree at most k such that y[i]=P(x[i]) for all but at most e values of i in the range 1 <= i <= n. We give an algorithm that solves this problem for e < n - sqrt{kn}, which improves over the previous best result due to Sudan [Journal of Complexity, Vol. 13, 1997], for every choice of k and n. Of particular interest is the case of k/n > 1/3, where the result yields the first asymptotic improvement since Peterson's original algorithm nearly four decades ago.The algorithm generalizes to solve the list decoding problem for other algebraic codes, specifically alternant codes (a class of codes including BCH codes) and algebraic-geometric codes. In both cases, we obtain a list decoding algorithm that corrects up to n - sqrt{n(n-d')} errors, where n is the block length and d' is the designed distance of the code. The improvement for the case of algebraic-geometric codes extends the methods of Shokrollahi and Wasserman [STOC '98] and improves upon their bound for every choice of n and d'. We also present some other consequences of our algorithm including a solution to a weighted curve fitting problem, which is of use in soft-decision decoding algorithms for Reed-Solomon codes.