Supersymmetrization: AKSZ and beyond?

Abstract: In this paper we describe multigraded generalizations of some constructions useful for mathematical understanding of gauge theories: we perform a near-at-hand generalization of the Aleksandrov--Kontsevich--Schwarz--Zaboronsky procedure, we also extend the formalism of $Q$-bundles introduced first by A. Kotov and T. Strobl. We compare these approaches studying some supersymmetric sigma models important in theoretical physics.