As an entry point to differential geometry, this chapter defines embedded submanifolds as subsets of linear spaces which can be locally defined by equations satisfying certain regularity conditions. Such sets can be linearized, yielding the notion of tangent space. The chapter further defines what it means for a map to and from a submanifold to be smooth, and how to differentiate such maps. The (disjoint) union of all tangent spaces forms the tangent bundle which is also a manifold. That makes it possible to define vector fields (maps which select a tangent vector at each point) and retractions (smooth maps which generate curves passing through any point with any given velocity). The chapter then proceeds to endow each tangent space with an inner product (turning each one into a Euclidean space). Under some regularity conditions, this extra structure turns the manifold into a Riemannian manifold. This makes it possible to define the Riemannian gradient of a real function. Taken together, these concepts are sufficient to build simple algorithms in the next chapter. An optional closing section defines local frames: They are useful for proofs but can be skipped for practical matters.