The fundamental trade-off in coding theory is the one between the rate of the code (a measure of amount of redundancy introduced) and the amount of errors that can be corrected. In this talk, I will describe an explicit construction of codes that achieves the optimal trade-off between these parameters, for a worst-case noise model where the channel can corrrupt the codeword arbitrarily subject to a bound on the total number of errors. Formally, for every 0 < R < 1 and eps > 0, we present an explicit construction of codes of rate R (which encode R.n symbols into n symbols) that can be list decoded in polynomial time from up to a fraction (1-R-eps) of errors. Clearly, one needs at least Rn correct symbols in the received word to have any hope of identifying the Rn message symbols, so recovering from (R+eps)n correct symbols is information-theoretically best possible. Our codes are simple to describe: they are certain *folded* Reed-Solomon codes, which are just Reed-Solomon codes, but viewed as a code over a larger alphabet by a careful bundling of codeword symbols. The talk will introduce the problem of list decoding, and then give a peek into the algebraic ideas and constructions that lead to the above result. Time permitting, I will also describe some challenging questions concerning algebraic-geometric codes that, if resolved, could improve the decoding complexity as one approaches the optimal trade-off. Based on joint work with Atri Rudra (UW).