  We define Hardy spaces $\mathcal{H}^p$ for quasiregular mappings in the
plane, and show that for a particular class of these mappings many of the
classical properties that hold in the classical setting of analytic mappings
still hold. This particular class of quasiregular mappings can be characterised
in terms of composition operators when the symbol is quasiconformal. Relations
between Carleson measures and Hardy spaces play an important role in the
discussion. This program was initiated and developed for Hardy spaces of
quasiconformal mappings by Astala and Koskela in 2011 in their paper
$\mathcal{H}^p$-theory for Quasiconformal Mappings.
