We study the $L^2$-Betti numbers of fiber bundles $F\rightarrow E\rightarrow B$ of manifolds. Under certain conditions (e.g., when $F$ is simply connected), $b^{(2)}_{\ast}(E)$ can be computed using the twisted $L^2$-Betti numbers of $B$. We relate the twisted and untwisted $L^2$-Betti numbers of $B$ when $\pi_1(B)$ is locally indicable. As an application, we compute $b^{(2)}_{\ast}(E)$ when $B$ is either a surface or a non-positively curved $3$-manifold. This is a joint work with Dawid Kielak.