What is Local Models Semantics?

In recent papers a new semantics, called Local Models Semantics, was presented and used to provide a foundation to reasoning with contexts. Local Models Semantics captures and makes precise the two main principles underlying contextual reasoning: the, so-called, Principle of Locality and Principle of Compatibility. In this chapter we aim at explaining the main intuitions underlying Local Models Semantics, its fundamental logical properties, and its relation with contextual reasoning. The emphasis is on motivations and intuitions, rather than on technicalities.