| Summary: | An {\omega}-language is a set of infinite words over a finite alphabet X. We
consider the class of recursive {\omega}-languages, i.e. the class of
{\omega}-languages accepted by Turing machines with a B\"uchi acceptance
condition, which is also the class {\Sigma}11 of (effective) analytic subsets
of X{\omega} for some finite alphabet X. We investigate here the notion of
ambiguity for recursive {\omega}-languages with regard to acceptance by B\"uchi
Turing machines. We first present in detail essentials on the literature on
{\omega}-languages accepted by Turing Machines. Then we give a complete and
broad view on the notion of ambiguity and unambiguity of B\"uchi Turing
machines and of the {\omega}-languages they accept. To obtain our new results,
we make use of results and methods of effective descriptive set theory.
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