This chapter classifies all one-dimensional smooth manifolds, showing that every connected smooth one-manifold is diffeomorphic to one of four basic examples. The proof employs parametrisation by arc length. Using this classification, we demonstrate that smooth manifolds with boundary do not admit a retraction onto their boundary. As an application, we prove Brouwer’s Fixed Point Theorem, first for smooth self-maps of a closed disc in Euclidean space, then for continuous maps, and show by example that the result fails for self-maps of an open ball. Exercises include using Brouwer’s theorem to prove the Perron—Frobenius theorem on the existence of non-negative eigenvalues for matrices with non-negative entries, and showing that the circle is not simply connected.