  In this article we explain how the coordinate ring of each (open) Schubert
variety in the Grassmannian can be identified with a cluster algebra, whose
combinatorial structure is encoded using (target labelings of) Postnikov's
plabic graphs. This result generalizes a theorem of Scott (Scott 2006) for the
Grassmannian, and proves a folklore conjecture for Schubert varieties that has
been believed by experts since (Scott 2006), though the statement was not
formally written down until M\"uller-Speyer explicitly conjectured it
(M\"uller-Speyer 2016). To prove this conjecture we use a result of Leclerc
(Leclerc 2016), who used the module category of the preprojective algebra to
prove that coordinate rings of many Richardson varieties in the complete flag
variety can be identified with cluster algebras. Our proof also uses a
construction of Karpman (Karpman 2016) to build plabic graphs associated to
reduced expressions. We additionally generalize our result to the setting of
skew Schubert varieties; the latter result uses generalized plabic graphs, i.e.
plabic graphs whose boundary vertices need not be labeled in cyclic order.
