  Given a nonunital $\mathrm{C}^*$-algebra $A$ one constructs its corona
algebra $\mathcal M(A)/A$. This is the noncommutative analog of the
\v{C}ech-Stone remainder of a topological space. We analyze the two faces of
these algebras: the first one is given assuming CH, and the other one arises
when Forcing Axioms are assumed. In their first face, corona
$\mathrm{C}^*$-algebras have a large group of automorphisms that includes
nondefinable ones. The second face is the Forcing Axiom one; here the
automorphism group of a corona $\mathrm{C}^*$-algebra is as rigid as possible,
including only definable elements
