  We define the Hardy spaces of free noncommutative functions on the
noncommutative polydisc and the noncommutative ball and study their basic
properties. Our technique combines the general methods of noncommutative
function theory and asymptotic formulae for integration over the unitary group.
The results are the first step in developing the general theory of free
noncommutative bounded symmetric domains on the one hand and in studying the
asymptotic free noncommutative analogues of classical spaces of analytic
functions on the other.
