Introduction

From metals to neutron stars, superconductors to nuclear plasmas, phases of matter described by Fermi surfaces and their instabilities are proliferous. The question “What are the different possible ways that interacting fermions can behave at macroscopic scales?” is as easy to pose as it is difficult to answer. The possibilities are endless and ever-growing and stand tall and sturdy as a counterpoint to the traditional reductionist-constructivist hypothesis in physics. We present in this chapter a brief historical survey of the various tools that have been developed to address this question for the restricted class of phases of matter that have a Fermi surface.