  In the paper we discuss three different notions of extremal holomorphic
mappings: weak $m$-extremals, $m$-extremals and $m$-complex geodesics. We
discuss relations between them in general case and in the special cases of unit
ball, classical Cartan domains, symmetrised bidisc and tetrablock. In
particular we show that weak $3$-extremal maps in the symmetrised bidisc are
rational thus giving the (partial) answer to a problem posed in a recent paper
by J. Agler, Z. Lykova and N. J. Young.
