This paper presents a combinatorial study of the super plactic monoid of type A, which is related to the representations of the general linear Lie superalgebra. We introduce the analogue of the Schützenberger's jeu de taquin on the structure of super tableaux over a signed alphabet. We show that this procedure which transforms super skew tableaux into super Young tableaux is compatible with the super plactic congruence and it is confluent. We deduce properties relating the super jeu de taquin to insertion algorithms on super tableaux. Moreover, we introduce the super evacuation procedure as an involution on super tableaux and we show its compatibility with the super plactic congruence. Finally, we describe the super jeu de taquin in terms of Fomin's growth diagrams in order to give a combinatorial version of the super Littlewood--Richardson rule.