In this paper we are concerned with two classes of conformally invariant spaces of analytic functions in the unit disc D, the Besov spaces Bp (1 <= p < inf) and the Qs spaces (0<s< inf). 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 Tg or the companion operator Ig with symbol g is a bounded operator from X into Y.
