Since Wiles proved (most of) the Taniyama-Shimura Conjecture in the 1990s many representations $\rho$ of the absolute Galois group of the rationals with values in $GL_n(\mathbf{Z}_p)$ were shown to be modular, i.e., to arise from automorphic forms. In this talk we will concentrate on recent progress on modularity of $\rho$ when its reduction mod $p$ is reducible and has three Jordan-Holder factors. We will discuss consequences of these results for the modularity of some abelian varieties of dimension >1. This is joint work with Tobias Berger.