Constraint maps arise as critical points of variational energies under image constraints and naturally generalize classical obstacle-type problems to the vectorial setting. In this talk, I will give a brief overview of recent progress in their analysis, with a focus on a new tool we developed—a quantitative unique continuation principle. This principle provides a powerful bridge between energy decay and nondegeneracy near the free boundary. I will explain how this insight allowed us to resolve a fundamental regularity issue for constraint maps near their free boundary. Time permitting, I will also discuss a few future directions concerning the parametric minimal surfaces and harmonic maps. The talk is based on recent joint works with Alessio Figalli (ETH), André Guerra (Cambridge), and Henrik Shahgholian (KTH).