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In this paper we provide a summary of the fundamental properties of probabilistic automata over infinite words. Such probabilistic automata are a variant of standard automata with Büchi or other *** -regular acceptance conditions, such as Rabin, Streett, parity or Müller, where the nondeterministic choices are resolved probabilistically. Acceptance of an infinite input word can be defined in different ways: by requiring that (i) almost all runs are accepting, or (ii) the probability for the accepting runs is positive, or (iii) the probability measure of the accepting runs is beyond a certain threshold. Surprisingly, even the qualitative criteria (i) and (ii) yield a different picture concerning expressiveness, efficiency, and decision problems compared to the nondeterministic case.