Moving boundary problems lie at heart of most nonequilibrium phase change processes in materials, but also in morphogenesis in biology. Their interest, as well as difficulty, lies in the intricate spatio-temporal patterns that are charateristic of their evolution. We consider in this set of lectures the fundamentals of a mesoscopic description of boundary moving problems, the principles required in the derivation of their transport models, and make a few remarks on computational methodologies to address them.