In this paper, we introduce the notion of super-immanants for supermatrices over a supercommutative algebra. Using the super Schur-Weyl duality we show that the super immanants play a significant role in covariant tensor representations of the general linear Lie superalgebra. Among various things, we obtain a supertrace formula for super-immanants, which generalizes Kostant's trace formula to the super setting. Furthermore, we show that the Littlewood correspondences between super-immanants and supersymmetric polynomials establish an isomorphism between their corresponding algebras.