The authors consider the weighted Dirichlet space \({\mathcal D}_\tau \) of the unit ball in \({\mathbb C}^n \). These spaces include the classical Dirichlet space (\(\tau=1 \)), the Hardy space (\(\tau=0 \)), and the Bergman space (\(\tau=-1 \)), and are defined in the paper in the standard way via their Taylor coefficients. A complex-valued function \(\phi \) defined on the unit ball is said to be a pointwise multiplier (or simply a multiplier) from \({\mathcal D}_\tau \) to \({\mathcal D}_\mu \) if \( f \in {\mathcal D}_\mu \) whenever \(f\in {\mathcal D}_\tau \). The multipliers between function spaces have been studied by a number of authors. In the case of Dirichlet type spaces, we refer the reader to the papers by \textit{D. Stegenga} [Ill. J. Math. 24, 113-139 (1980; Zbl 0432.30016)] and \textit{R. Kerman} and \textit{E. Sawyer} [Trans. Am. Math. Soc. 309, 87-98 (1988; Zbl 0657.31019)], for example, where this concept was studied from the viewpoint of Carleson measures (there are also several related papers by Axler, Shields, and others). In the paper under review, the authors obtain complete characterizations of the multipliers from \({\mathcal D}_\tau \) into \({\mathcal D}_\mu \) for a certain range of indices, thus generalizing the known facts about the functions of one complex variable. In other cases, they construct explicit examples showing that functions in certain spaces cannot be multipliers. Their proofs are based on the Cauchy-Schwarz inequality, Stirlings formula, and manipulations of power series. The References contain several misprints.