  In this paper we present results concerning orthogonality in Hilbert
$C^*$-modules. Moreover, for a $C^*$-algebra $\mathscr{A}$, we prove theorems
concerning the multi-$\mathscr{A}$-linearity and its preservation by
$\mathscr{A}$-linear operators. New version of solution of the orthogonality
equation on Hilbert $C^*$-modules and mappings preserving orthogonality are
also investigated.
