The main object of the paper under review is the wonderful completion \(X\) of a symmetric variety \(G/H\) constructed by C. De Concini and C. Procesi [in: Invariant theory, Lect. Notes Math. 996, 1–44 (1983; Zbl 0581.14041)] (\(G\) is an adjoint semisimple group, \(H\) is the fixed subgroup of an involutive automorphism of \(G\)). The authors study the ring of sections \(A(X):=\bigoplus_{L\in \text{Pic}(X)}H^0(X,L)\) and construct a standard monomial theory for this ring. This can be viewed as a generalization of Littelman’s theory (where \(H=P\) is a parabolic subgroup and no completion is needed). As an application, the authors construct a degeneration of \(A(X)\) to the coordinate ring of the product of a multicone over a flag variety and an affine space (that, in particular, implies that \(A(X)\) has rational singularities). Some of intermediate results are interesting by their own, for example, an explicit description of a basis of \(\text{Pic}(X)\) in terms of spherical weights and a generalization of the Parthasarathy-Ranga Rao-Varadarajan conjecture [K. R. Parthasarathy, R. Ranga Rao and V. S. Varadarajan, Bull. Am. Math. Soc. 72, 522–525 (1966; Zbl 0178.16802)].