We discuss joint work with Ying Hu concerning the left-orderability of the fundamental groups of cyclic branched covers of links which admit co-oriented taut foliations. In particular, we do this for cyclic branched covers of fibered knots in integer homology $3$-spheres and cyclic branched covers of closed braids in the $3$-sphere. The latter allows us to complete the proof of the L-space conjecture for closed, connected, orientable, irreducible $3$-manifolds containing a genus $1$ fibered knot. We also prove that the universal abelian cover of a manifold obtained by generic Dehn surgery on a hyperbolic fibered knot in an integer homology $3$-sphere admits a co-oriented taut foliation and has left-orderable fundamental group, even if the surgered manifold does not.