Gauge theory excels at understanding minimal genus problems for 3- and 4-manifolds.  A notable triumph is its resolution of the symplectic Thom conjecture, asserting that the genus of a symplectic surface in a closed symplectic 4-manifold is no larger than any smooth surface homologous to it.  One can formulate versions of this conjecture for surfaces with boundary lying in a 3-manifold, and I'll discuss work in progress with Katherine Raoux which, curiously, seem in certain situations to extend these relative Thom conjectures outside the symplectic (or complex) realm.