The self-dual U(1)-Yang-Mills-Higgs functionals are a natural family of energies associated to sections and metric connections of Hermitian line bundles, whose critical points (particularly in the 2-dimensional and Kaehler settings) are objects of long-standing interest in low-dimensional gauge theory. In this talk, we will discuss joint work with Alessandro Pigati characterizing the behavior of critical points over manifolds of arbitrary dimension. We show in particular that critical points give rise to minimal submanifolds of codimension two in certain natural scaling limits, and use this information to provide new constructions of codimension-two minimal varieties in arbitrary Riemannian manifolds. We will also discuss recent work with Davide Parise and Alessandro Pigati developing the associated Gamma-convergence machinery, and describe some geometric applications.