  A hallmark of the phase diagrams of quantum materials is the existence of
multiple electronic ordered states, which, in many cases, are not independent
competing phases, but instead display a complex intertwinement. In this review,
we focus on a particular realization of intertwined orders: a primary phase
characterized by a multi-component order parameter and a fluctuation-driven
vestigial phase characterized by a composite order parameter. This concept has
been widely employed to elucidate nematicity in iron-based and cuprate
superconductors. Here we present a group-theoretical framework that extends
this notion to a variety of phases, providing a classification of vestigial
orders of unconventional superconductors and density-waves. Electronic states
with scalar and vector chiral order, spin-nematic order, Ising-nematic order,
time-reversal symmetry-breaking order, and algebraic vestigial order emerge
from one underlying principle. The formalism provides a framework to understand
the complexity of quantum materials based on symmetry, largely without
resorting to microscopic models.
