In order to liquidate a large position in an asset, investors face a tradeoff between price volatility and market impact. The classical approach to this problem is to model volatility via a Brownian motion, and separate price impact into its permanent and temporary components. In this thesis, we consider two variations of the Chriss-Almgren model for temporary price impact. The first model investigates the infinite-horizon optimal liquidation problem in a market with float-dependent, nonlinear temporary price impact. The value function of the investor’s basket and the optimal strategy are characterized in terms of classical solutions of nonlinear parabolic partial differential equations. Depending on the price impact parameters, liquidation may require finite or infinite time. The second model considers time-varying market depth, in that intense trading increases temporary price-impact, which otherwise reverts to a long-term level. We find the optimal execution policy in a finite horizon for an investor with constant risk aversion, and derive the solution using calculus of variation techniques. Although the model potentially allows for price manipulation strategies, these policies are never optimal. We study the non time-constrained case as a limit to the finite-horizon case and explain the solution through a quasi-linear PDE.