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 Büchi acceptance condition, which is also the class $\Sigma_1^1$ 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 Büchi 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 Büchi 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.