I will discuss recent results on the bifurcation of spatially localized states from a continuum of extended states. This phenomenon plays an important role in the mathematical study of wave propagation in ordered microstructures, which are perturbed by spatially compact or non-compact defects. Near the bifurcation point, there is strong spatial scale separation and one expects the "natural'' homogenized equation to govern. Our first example is one in which this intuition does not apply, and an appropriate effective equation must be derived. (Joint work with V. Duchene and I. Vukicevic) Our second example concerns the bifurcation of “topologically protected edge states” in a class of periodic structures, perturbed by a (non-compact) domain wall. Such states play a central role in many recently studied systems in condensed matter physics and photonics. (Joint work with C.L. Fefferman and J.P. Lee-Thorp)