For a cardinal \(\kappa\), a space \(X=(X,{\mathcal T})\) is said to be \(\kappa\)-resolvable if \({\mathcal P}(X)\) contains \(\kappa\)-many pairwise disjoint \(\mathcal T\)-dense sets; in this terminology, \(X\) is resolvable if \(X\) is 2-resolvable, and \(X\) is maximally resolvable if \(X\) is \(\Delta(X)\)-resolvable, where \(\Delta(X)=\min\{|U|:\emptyset \neq U \in {\mathcal T}\}\). In the positive direction, the author shows that spaces of the following kinds are resolvable: (P1) every analytic subset \(X\) with \(\Delta(X) > \omega\) in a regular, countably compact space; (P2) every regular, \(\sigma\)-compact space \(X\) with \(\Delta(X) > \omega\); and (P3) every \(CA\)-set, and every \(PCA\)-set, in a compact Hausdorff space. In the negative direction, he presents examples showing that spaces of the following kinds need not be resolvable: (N1) a countably compact, Hausdorff space; (N2) a \(\sigma\)-compact Hausdorff space \(X\) with \(\Delta(X) > \omega\); and (N3) a space of the form \(X = X_1\cap X_2\) with \(\Delta(X) >\omega\), \(X_1\) analytic in a Tikhonov space \(Y\) and \(X_2\) a \(CA\)-set in \(Y\). The author remarks that a different example as in (N2) was given earlier by O. Pavlov. That example and the author's have interest in view of the fact that every Tikhonov space without isolated points is \(\omega\)-resolvable. The above-cited results are all ZFC. Among the questions designated by the author as unsolved are these: (Q1) Are projective sets of order \(\geq 4\) in compacta necessarily resolvable? (Q2) are the spaces \(X\) in (P3) necessarily maximally resolvable?