  Provability logic concerns the study of modality $\Box$ as provability in
formal systems such as Peano arithmetic. Natural, albeit quite surprising,
topological interpretation of provability logic has been found in the 1970's by
Harold Simmons and Leo Esakia. They have observed that the dual $\Diamond$
modality, corresponding to consistency in the context of formal arithmetic, has
all the basic properties of the topological derivative operator acting on a
scattered space. The topic has become a long-term project for the Georgian
school of logic led by Esakia, with occasional contributions from elsewhere.
  More recently, a new impetus came from the study of polymodal provability
logic GLP that was known to be Kripke incomplete and, in general, to have a
more complicated behavior than its unimodal counterpart. Topological semantics
provided a better alternative to Kripke models in the sense that GLP was shown
to be topologically complete. At the same time, new fascinating connections
with set theory and large cardinals have emerged.
  We give a survey of the results on topological semantics of provability logic
starting from first contributions by Esakia. However, a special emphasis is put
on the recent work on topological models of polymodal provability logic. We
also included a few results that have not been published so far, most notably
the results of Section 6 (due the second author) and Sections 10, 11 (due to
the first author).
