In the recent decades, a new frontier has emerged with a similar goal, that is to con-trol the optical properties of materials. If we can engineer materials that prohibit or allow the propagation of light at certain frequencies, or localize it, our technology would benefit greatly. For instance, one envisions optical integrated circuits, which potentially can op- erate faster than our current semiconductor ones. The materials that could lead us to such goal are the so called photonic crystals.Sections B and C of contain an exposition and the proofs of our results concerning existence and confinement of guided waves in photonic crystal waveguides. Namely, we prove existence and confinement of guided waves through a linear defect in a PBG material, provided some ―strength‖ conditions on the defect. The results are obtained both for the scalar (corresponding to acoustic or 2D photonic guides) and the full 3D Maxwell cases. In the last couple of decades, engineers have been able to produce thin, graph-like structures (quantum wires, mesoscopic systems). The need to study propagation of waves in such structures has lead to the birth of quantumgraph theory.