  A homogeneous quasimorphism $\phi$ on a normal subgroup $N$ of $G$ is said to
be $G$-invariant if $\phi(gxg^{-1}) = \phi(x)$ for every $g \in G$ and for
every $x \in N$. Invariant quasimorphisms have naturally appeared in symplectic
geometry and the extension problem of quasimorphisms. Moreover, it is known
that the existence of non-extendable invariant quasimorphisms is closely
related to the behavior of the stable mixed commutator length
$\mathrm{scl}_{G,N}$, which is a certain generalization of the stable
commutator length $\mathrm{scl}_G$.
  In this survey, we review the history and recent developments of invariant
quasimorphisms and stable mixed commutator length. The topics we treat include
several examples of invariant quasimorphisms, Bavard's duality theorem for
invariant quasimorphisms, Aut-invariant quasimorphisms, and the estimation of
the dimension of spaces of non-extendable quasimorphisms. We also mention the
extension problem of partial quasimorphisms.
