This chapter sets up a formal framework for local search and provides a certain number of useful definitions for the rest of this work. It gives a few examples of combinatorial optimization problems with the neighborhoods classically used. The chapter devotes the complexity class polynomial-time local search (PLS). It examines in more detail the complexity of the standard local search algorithm. On the subject of the quality of the solutions, the chapter looks at a restrictive theorem regarding the quality of the solutions provided by a local search algorithm for NP-hard and strongly NP-hard problems. It describes several classical combinatorial optimization problems for which local search algorithms provide solutions that are (1 + ε{lunate})-approximate with regard to a global optimum. Furthermore, despite its use in solving NP-hard problems, local search is also used in other domains such as game theory. © ISTE Ltd 2014. All rights reserved.