We show that the natural operation of connected sum for graphs can be used to prove at once most of the universality results from the literature concerning graph homomorphism. In doing so, we significantly improve many existing theorems and solve some natural open problems. Despite its simplicity, our technique unexpectedly leads to applications in quite diverse areas of mathematics, such as category theory, combinatorics, classical descriptive set theory, generalized descriptive set theory, model theory, and theoretical computer science. (Joint work with S. Scamperti)