We present 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 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 a super version of the Robinson—Schensted—Knuth correspondence for super tableaux and we give a combinatorial version of the super Littlewood--Richardson rule, which describes the multiplicity of a super Schur polynomial in a product of super Schur polynomials.