  This paper seeks to extend the theory of composition operators on analytic
functional Hilbert spaces from analytic symbols to quasiconformal ones. The
focus is the boundedness but operator-theoretic questions are discussed as
well. In particular, we present a thorough analysis of $L^p$-estimates of a
class of singular integral operators $P_\varphi$ associated with a
quasiconformal mapping $\varphi$.
