  We survey results on mutations of Jacobian algebras, while simultaneously
extending them to the more general setup of frozen Jacobian algebras, which
arise naturally from dimer models with boundary and in the context of the
additive categorification of cluster algebras with frozen variables via
Frobenius categories. As an application, we show that the mutation of
cluster-tilting objects in various such categorifications, such as the
Grassmannian cluster categories of Jensen-King-Su, is compatible with
Fomin-Zelevinsky mutation of quivers. We also describe an extension of this
combinatorial mutation rule allowing for arrows between frozen vertices, which
the quivers arising from categorifications and dimer models typically have.
