For a given class T of compact Hausdorff spaces, let Y(T) denote the class of ℓ-groups G such that for each g∈G, the Yosida space Y(g) of g belongs to T. Conversely, if R is a class of ℓ;-groups, then T(R) stands for the class of all spaces which are homeomorphic to a Y(g) for some g∈G∈R. The correspondences T↦Y(T) and R↦T(R) are examined with regard to several closure properties of classes. Several sections are devoted to radical classes of ℓ-groups whose Yosida spaces are zero-dimensional. There is a thorough discussion of hyper-projectable ℓ-groups, followed by presentations on Y(e.d.), where e.d. denotes the class of compact extremally disconnected spaces, and, for each regular uncountable cardinal κ, the class Y(discκ), where discκ stands for the class of all compact κ-disconnected spaces. Sample results follow. Every strongly projectable ℓ-group lies in Y(e.d.). The ℓ-group G lies in Y(e.d.) if and only if for each g∈GY(g) is zero-dimensional and the Boolean algebra of components of g, comp(g), is complete. Corresponding results hold for Y(discκ). Finally, there is a discussion of Y(F), with F standing for the class of compact F-spaces. It is shown that an Archimedean ℓ-group G is in Y(F) if and only if, for each pair of disjoint countably generated polars P and Q, G=P⊥+Q⊥.