  We study the decidability and expressiveness issues of $\mu$-calculus on data
words and data $\omega$-words. It is shown that the full logic as well as the
fragment which uses only the least fixpoints are undecidable, while the
fragment containing only greatest fixpoints is decidable. Two subclasses,
namely BMA and BR, obtained by limiting the compositions of formulas and their
automata characterizations are exhibited. Furthermore, Data-LTL and
two-variable first-order logic are expressed as unary alternation-free fragment
of BMA. Finally basic inclusions of the fragments are discussed.
