We introduce and study the set of radical semistar operations of an integral domain D. We show that their set is a complete lattice that is the join-completion of the set of spectral semistar operations, and we characterize when every radical operation is spectral (under the hypothesis that D is rad-colon coherent). When D is a Prüfer domain such that every set of minimal prime ideals is scattered, we completely classify stable semistar operations.
