  In this mostly expository paper, we present recent progress on infinite
(weak) cluster categories that are related to triangulations of the disk, with
and without a puncture. First we recall the notion of a cluster category. Then
we move to the infinite setting and survey recent work on infinite cluster
categories of types $\mathbb{A}$ and $\mathbb{D}$. We conclude with our
contributions, two infinite families of infinite (weak) cluster categories of
type $\mathbb{D}$. We first present a discrete, infinite version of Schiffler's
combinatorial model of the punctured disk with marked points. We then produce
each (weak) cluster category starting with representations of thread quivers,
taking the derived category, and then taking the appropriate orbit category. We
show that the combinatorics in the (weak) cluster categories match with the
corresponding combinatorics of the punctured disk with countably-many marked
points. We also state two conjectures concerning weak cluster structures inside
our (weak) cluster categories.
