  We give a unified overview of the study of the effects of additional set
theoretic axioms on quotient structures. Our focus is on rigidity, measured in
terms of existence (or rather non-existence) of suitably \emph{non-trivial}
automorphisms of the quotients in question. A textbook example for the study of
this topic is the Boolean algebra $\mathcal P(\mathbb N)/Fin$, whose behavior
is the template around which this survey revolves: Forcing axioms imply that
all of its automorphisms are \emph{trivial}, in the sense that they are induced
by almost permutations of $\mathbb N$, while under the Continuum Hypothesis
this rigidity fails and $\mathcal P(\mathbb N)/Fin$ admits uncountably many
non-trivial automorphisms. We consider far-reaching generalizations of this
phenomenon and present a wide variety of situations where analogous patterns
persist, focusing mainly (but not exclusively) on the categories of Boolean
algebras, \v Cech--Stone remainders, and $\mathrm{C}^*$-algebras. We survey the
state of the art and the future prospects of this field, discussing the major
open problems and outlining the main ideas of the proofs whenever possible.
