Let G be an r-partite graph such that the edge density between any two parts is at least α. How large does α need to be to guarantee that G contains a connected transversal, that is, a tree on r vertices meeting each part in one vertex? And what if instead we want to guarantee the existence of a Hamiltonian transversal? In this paper we initiate the study of such extremal multipartite graph problems, obtaining a number of results and providing many new constructions, conjectures and further questions.