  This survey provides a self-contained account of $M$-estimation of
multivariate scatter. In particular, we present new proofs for existence of the
underlying $M$-functionals and discuss their weak continuity and
differentiability. This is done in a rather general framework with
matrix-valued random variables. By doing so we reveal a connection between
Tyler's (1987) $M$-functional of scatter and the estimation of proportional
covariance matrices. Moreover, this general framework allows us to treat a new
class of scatter estimators, based on symmetrizations of arbitrary order.
Finally these results are applied to $M$-estimation of multivariate location
and scatter via multivariate $t$-distributions.
