Classical Calculus of Variations problems were motivated by analytical mechanics and geometrical optics, where the unknown was a curve in phase space. The problem of multiple integrals in Calculus of Variations is motivated by non-linear elasticity and martensitic phase transitions. In the latter setting, the presence of phase boundaries is essential for understanding the unusual behavior of shape memory alloys. In the variational principles used in these models the unknown is a vector field rather than a trajectory. Surprisingly, the generalization from scalar variational problems to vectorial ones turned out to be anything but straightforward. I will discuss my recent work in this area on smooth local minimizers (joint with Tadele Mengesha) and local minimizers with phase boundaries (joint with Vladislav Kucher and Lev Truskinovsky).