This talk discusses the relationship between patching (building a global object by building it locally) and deformation theory, in the context of projective curves. Several views of patching will be presented, along with the connection to Cartan's Lemma and Grothendieck's Existence Theorem. Applications will be described to deformations of modules and algebras, and to the inverse Galois problem. (A more recent application to differential Galois theory, due to the speaker and Julia Hartmann, will be presented in the second talk later today.)