  Quantum finite automata (QFAs) have been extensively studied in the
literature. In this paper, we define and systematically study quantum B\"uchi
automata (QBAs) over infinite words to model the long-term behavior of quantum
systems, which extend QFAs. We introduce the classes of $\omega$-languages
recognized by QBAs in probable, almost sure, strict and non-strict threshold
semantics. Several pumping lemmas and closure properties for QBAs are proved.
Some decision problems for QBAs are investigated. In particular, we show that
there are surprisingly only at most four substantially different classes of
$\omega$-languages recognized by QBAs (out of uncountably infinite). The
relationship between classical $\omega$-languages and QBAs is clarified using
our pumping lemmas. We also find an $\omega$-language recognized by QBAs under
the almost sure semantics, which is not $\omega$-context-free.
