  We construct six-dimensional (6D) F-theory models in which discrete
$\mathbb{Z}_5, \mathbb{Z}_4, \mathbb{Z}_3,$ and $\mathbb{Z}_2$ gauge symmetries
arise. We demonstrate that a special family of "Fano 3-folds" is a useful tool
for constructing the aforementioned models. The geometry of Fano 3-folds in the
constructions of models can be useful for understanding discrete gauge
symmetries in 6D F-theory compactifications. We argue that the constructions of
the aforementioned models are applicable to Calabi-Yau genus-one fibrations
over any base space, except models with a discrete $\mathbb{Z}_5$ gauge group.
We construct 6D F-theory models with a discrete $\mathbb{Z}_5$ gauge group over
the del Pezzo surfaces, as well as over $\mathbb{P}^1\times\mathbb{P}^1$ and
$\mathbb{P}^2$. We also discuss some applications to four-dimensional F-theory
models with discrete gauge symmetries.
