Abstract: Recent years have seen a rising interest in invariant geometric structures and flows on $n$-dimensional almost Abelian solvmanifolds $M=\Gamma\backslash G$. This interest stems from the fact that such compact manifolds $M$ and invariant geometric structures on them have provided important (counter-)examples and are easily accesible since $G$ is fully determined by one endomorphim $f$ of $\mathbb{R}^{n-1}$. Many of the studied invariant geometric structures may be described as torsion-free $H$-structures for a specific linear Lie group $H$, and for these particular $H$s characterisations of the endomorphisms $f$ for which the associated almost Abelian Lie group $G$ admits a left-invariant torsion-free $H$-structure have been obtained. In my talk, I describe a framework which allows, for an arbitrary linear Lie group $H$, to characterise those almost Abelian Lie group $G$ which admit a left-invariant torsion-free $H$-structure. The framework then allows to vastly extend the existing characterisations for single linear Lie groups $H$ to big families of such $H$s.