In this thesis we prove homological stability for both general linear groups of modules over a ring with finite stable rank and unitary groups of quadratic modules over a ring with finite unitary stable rank. In particular, we do not assume the modules and quadratic modules to be well-behaved in any sense: for example, the quadratic form may be singular. This extends results by van der Kallen and Mirzaii--van der Kallen respectively. Combining these results with the machinery introduced by Galatius--Randal-Williams to prove homological stability for moduli spaces of simply-connected manifolds of dimension $2n\ge 6$, we get an extension of their result to the case of virtually polycyclic fundamental groups. We also prove the corresponding result for manifolds equipped with tangential structures.

A result on the stable homology groups of moduli spaces of manifolds by Galatius--Randal-Williams enables us to make new computations using our homological stability results. In particular, we compute the abelianisation of the mapping class groups of certain $6$-dimensional manifolds. The first computation considers a manifold built from $RP6$ which involves a partial computation of the Adams spectral sequence of the spectrum $MT$Pin$\mathrm{<mrow\; data-mjx-texclass="ORD">-</mrow>}(6)$. For the second computation we consider Spin $6$-manifolds with $\pi 1\cong Z/2kZ$ and $\pi 2=0$, where the main new ingredient is an~analysis of the Atiyah--Hirzebruch spectral sequence for $MT\mathrm{Spin}(6)\wedge \Sigma \infty BZ/2kZ+$. Finally, we consider the similar manifolds with more general fundamental groups $G$, where $K1(Q[G\mathrm{ab}])$ plays a role.