  In this paper we are concerned with two classes of conformally invariant
spaces of analytic functions in the unit disc $\D$, the Besov spaces $B^p$
$(1\le p<\infty )$ and the $Q_s$ spaces $(0<s<\infty )$. Our main objective is
to characterize for a given pair $(X, Y)$ of spaces in these classes, the space
of pointwise multipliers $M(X, Y)$, as well as to study the related questions
of obtaining characterizations of those $g$ analytic in $\D $ such that the
Volterra operator $T_g$ or the companion operator $I_g$ with symbol $g$ is a
bounded operator from $X$ into $Y$.
