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This paper makes a comprehensive survey of results relating to finite Reesindex for semigroups. In particular, we survey of the state of knowledge onwhether various finiteness properties (such as finite generation, finitepresentability, automaticity, and hopficity) are inherited by finite Rees indexsubsemigroups and extensions. We survey other properties that are invariantunder passing to finite Rees index subsemigroups and extensions, such as thecofinality and number of ends. We prove some new results: inheritance ofword-hyperbolicity by finite Rees index subsemigroups, and inheritance of(geometric) hyperbolicity by finite Rees index extensions and subsemigroupswithin the class of monoids of finite geometric type. We also give someimproved counterexamples. All the results are summarized in a table.

For a few elements more: A survey of finite Rees index