This chapter introduces smooth maps between subsets of Euclidean space, building on the familiar notion of differentiable real-valued functions. We then present local parametrisations and define smooth manifolds embedded in Euclidean space. Smooth maps between manifolds are easily described in this concrete framework, and we examine basic examples such as spheres and the torus. Next, we introduce tangent spaces and the derivative of smooth maps, two fundamental tools in differential topology. The chapter concludes with an extended discussion of the tangent bundle as a smooth manifold. Exercises provide further examples, including the stereographic projection on spheres and an alternative description of tangent vectors.