This chapter introduces transversality, a key concept in differential topology that describes how a smooth map interacts locally with a given submanifold. The characterisation is made precise using tangent spaces and derivatives. We generalise the Preimage Theorem to transverse intersections and then specialise to intersections of submanifolds. Several examples are explored, including Brieskorn manifolds. As an application, we prove Sard’s Theorem on the existence of regular values, a fundamental result in differential topology. Building on Sard’s Theorem, we study Morse functions and establish Morse’s Theorem on the existence of such functions on smooth manifolds.