This chapter studies the local behaviour of smooth maps through the derivative. We begin with local diffeomorphisms and prove the Inverse Function Theorem, first for smooth maps between Euclidean spaces and then for maps between manifolds. Concrete examples illustrate the difference between local and global diffeomorphisms. Next, we consider maps with injective derivatives and introduce immersions, proving the Local Immersion Theorem, which gives immersions a particularly simple local description. Several examples lead to the concept of an embedding, and we examine the difference between immersions and embeddings before proving the Embedding Theorem. Exercises include, among others, a proof of the first part of the Constant Rank Theorem.