  We present a collection of results on (weak) $m$-extremals and $m$-geodesics,
concerning general properties, the planar case, quasi-balanced pseudoconvex
domains, complex ellipsoids, the Euclidean ball and boundary properties. We
prove $3$-geodesity of $3$-extremals in the Euclidean ball. Equivalence of weak
$m$-extremality and $m$-extremality in some class of convex complex ellipsoids,
containing symmetric ones and $\mathcal C^2$-smooth ones is showed. Moreover,
first examples of $3$-extremals being not $3$-geodesics in convex domains are
given.
