Historically, engineers have tried to avoid working with materials and structures under conditions where instabilities are likely to occur. Classical stability analyses have focused on predicting the onset of instability for use as an upper bound on allowable loads or as a design constraint. More recently it is becoming common to take advantage of these instabilities in order to design materials and structures with new and improved properties. Examples include, the remarkable properties and applications of shape memory alloys, phase transforming materials for solid state computer memory, and flexible high aspect-ratio airplane wings (providing improved manoeuvrability) designed to operate under flutter conditions and actively controlled against dynamic instability. Physical models (of materials, structures, aircraft, etc.) capable of predicting such instabilities are highly nonlinear. Thus, it is often extremely difficult to explore and understand all of the behavior predicted by a model. This presentation will review the theory and numerical implementation of Branch-Following and Bifurcation (BFB) techniques for exploring and understanding instabilities in physical systems. These techniques provide a systematic approach to the identification and interpretation of a model's behavior. The application of these techniques will be illustrated through examples: (i) atomistic modeling of shape memory alloys; (ii) finite element modeling of periodic structural materials such as honeycombs; and (iii) atomistic modeling of nano-structures such as nano-pillars.