Radical classes of lattice-ordered groups (\(\ell\)-groups) have been the subject of many researchers in recent years most notably the present author. Here, an important extension of these ideas is introduced in the study of radical classes of directed interpolation groups. [See \textit{L. Fuchs}, Ann. Sc. Norm. Sup. Pisa, Cl. Sci. Fis. Mat., III. Ser. 19, 1--34 (1965; Zbl 0125.28703); and \textit{K. R. Goodearl}, Partially ordered Abelian groups with interpolation. Providence, R.I.: AMS (1986; Zbl 0589.06008)]. These new radical classes satisfy closure under isomorphism, containment of convex directed subgroups, and closure under joins of convex directed subgroups. Let \({\mathfrak R}(G)\) denote the collection of all radical classes of \(\ell\)-groups and \({\mathfrak R}(I)\) denote the collection of all radical classes of directed interpolation groups. \({\mathfrak R}(G)\) fails to be a subcollection of \({\mathfrak R}(I)\). If \(G_1\) is any class of archimedean \(\ell\)-groups, then the radical class \(R(G_1)\) generated by \(G_1\) belongs to \({\mathfrak R}(I)\). This does not hold in general for non-archimedean \(\ell\)-groups. If \(A\in{\mathfrak R}(I)\), and \(\{B_j: j\in J\}\subseteq{\mathfrak R}(I)\), then \(A\land(\bigwedge_{j\in J} B_j)=\bigwedge_{j\in J}(A\land B_j)\). There exists an infinite injective mapping of the class of infinite cardinals into the class of all atoms of \({\mathfrak R}(I)\). From the results of \textit{W. C. Holland} [Czech. Math. J. 29, 11--12 (1979; Zbl 0432.06011)] it follows that each variety of \(\ell\)-groups belongs to \({\mathfrak R}(G)\). In particular the class of all Abelian \(\ell\)-groups belongs to \({\mathfrak R}(G)\). The case of interpolation groups is essentially different for it is proved that the class of all Abelian interpolation groups does not belong to \({\mathfrak R}(I)\). The following two questions of Goodearl are answered negatively: (A) Is every directed group with countable interpolation unperforated? (B) Is every directed group with countable interpolation isomorphic to a quotient group of a monotone \(\sigma\)-complete dimension group?