equivariance properties. Remarks on algorithmic aspects are included. The short Section \[sec-mvmedf\] points out some specific aspects of median filtering of multivariate images, focussing on the proper handling of different dimensionality of image domains and data ranges. Section \[sec-mvmedpde\] gives an overview of the results on approximations of PDEs by multivariate median filters that have been achieved so far, including one result the proof of which is still awaiting (and which is stated as conjecture therefore). The paper ends with a short summary in Section \[sec-sum\].

**Notations.**

Let us briefly introduce a few notations that will be used throughout this paper. By $\mathbb{R}$ we denote the set of real numbers. The symbol $\mathbb{R}^+$ stands for positive real numbers whereas $\mathbb{R}^+_0$ names nonnegative real numbers. As usual, $\mathbb{R}^n$ is the $n$-dimensional real vector or affine space which will sometimes be equipped with the standard Euclidean structure. In this case, $\lVert\boldsymbol{v}\rVert$ is the Euclidean norm of a vector $\boldsymbol{v}\in\mathbb{R}^n$, and $\langle \boldsymbol{v},\boldsymbol{w}\rangle$ the scalar product of vectors $\boldsymbol{v},\boldsymbol{w}\in\mathbb{R}^n$. By $\mathbb{Z}$ we denote the set of integers.

A *multiset* is a set with (integer) multiplicities; standard notations for sets, including union and intersection, are used for multisets in an intuitive way. The cardinality 