  We study Hardy spaces $\mathcal{H}^p$, $0<p<\infty$ for quasiregular mappings
on the unit ball $B$ in $\mathbb{R}^n$ which satisfy appropriate growth and
multiplicity conditions. Under these conditions we recover several classical
results for analytic functions and quasiconformal mappings in $\mathcal{H}^p$.
In particular, we characterize $\mathcal{H}^p$ in terms of non-tangential limit
functions and non-tangential maximal functions of quasiregular mappings. Among
applications we show that every quasiregular map in our class belongs to
$\mathcal{H}^p$ for some $p=p(n,K)$. Moreover, we provide characterization of
Carleson measures on $B$ via integral inequalities for quasiregular mappings on
$B$. We also discuss the Bergman spaces of quasiregular mappings and their
relations to $\mathcal{H}^p$ spaces and analyze correspondence between results
for $\mathcal{H}^p$ spaces and $\mathcal{A}$-harmonic functions.
  A key difference between the previously known results for quasiconformal
mappings in $\mathbb{R}^n$ and our setting is the role of multiplicity
conditions and the growth of mappings that need not be injective.
  Our paper extends results by Astala and Koskela, Jerison and Weitsman, Jones,
Nolder, and Zinsmeister.
