In this chapter, we analyze the evolution of the academic literature on optimal execution beyond the framework proposed by Almgren and Chriss. We discuss more general market impact models, the use of limit orders and dark pools, etc. We also focus on the modern understanding of the Almgren-Chriss model as a model for optimally scheduling the execution process, not as a model to execute Implementation Shortfall orders with market orders only. In particular, we discuss the two-layer approach of most execution algorithms, and the way the recent literature on optimal execution can help to design execution algorithms/tactics aimed at following the optimal trading curves computed in models a` la Almgren-Chriss. Furthermore, we discuss market impact estimation. In particular, we highlight the fact that market impact and execution cost estimates used in execution algorithms should be based on execution data rather than market data.