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This thesis studies the representation theory and linear structures of$\mathcal{Q}$-manifolds and higher Lie algebroids. We introduce differentialgraded modules (or for short DG-modules) of $\mathcal{Q}$-manifolds and theequivalent notion of representations up to homotopy in the case of Lie$n$-algebroids ($n\in\mathbb{N}$), as generalisations of the homonymousstructures that exist already in the case of ordinary Lie algebroids. Theadjoint and coadjoint modules are described, and the corresponding splitversions of the adjoint and coadjoint representations up to homotopy of Lie$n$-algebroids are explained. The compatibility of a graded Poisson bracketwith the homological vector field on a $\mathbb{Z}$-graded manifold is shown tobe equivalent to an (anti-)morphism from the coadjoint module to the adjointmodule, leading to an alternative characterisation of non-degeneracy of gradedPoisson structures. The Weil algebra of a general $\mathcal{Q}$-manifold isdefined and is computed explicitly in the case of Lie $n$-algebroids over abase (smooth) manifold $M$ together with a choice of a splitting and linear$TM$-connections. In addition, we study linear structures on$\mathbb{Z}$-graded manifolds, for which we see the connection with DG-modulesand representations up to homotopy. In the world of split Lie $n$-algebroids,this leads to the notion of VB-Lie $n$-algebroids. We prove that there is anequivalence between the category of VB-Lie $n$-algebroids over a Lie$n$-algebroid $\underline{A}$ and the category of $(n+1)$-term representationsup to homotopy of $\underline{A}$, generalising thus a well-known result fromthe theory of ordinary VB-algebroids.

$\mathbb{Z}$-graded supergeometry: Differential graded modules, higher algebroid representations, and linear structures