In the trading of large portfolios, the volatility risk eliminated by rapidly completing the trade program must be balanced against the market impact costs incurred when rapid execution is demanded. In previous work by R Almgren and N Chriss, explicit solutions were given for the case in which per-share impact costs are linear in trading rate. In this work we consider two nonlinear extensions. First, we obtain explicit solutions in the case that market impact depends nonlinearly on trading rate, including the popular square-root law. Second, we consider the case in which rapid trading increases not only the expected value of cost but also its uncertainty; although we do not obtain fully explicit solutions, we are able to give a complete asymptotic description and extract conclusions for practical trading.