An approach to the investigation of molecular structures in disordered solids, using two-dimensional (2D) nuclear magnetic resonance (NMR) exchange spectroscopy with magic angle spinning (MAS)I is described. This approach permits the determination of the relative orientation of two isotopically labeled chemical groups within a molecule in an unoriented sample, thus placing strong constraints on the molecular conformation. Structural information is contained in the amplitudes of crosspeaks in rotor-synchronized 2D MAS exchange spectra that connect spinning sideband Lines of the two labeled sites. The theory for calculating the amplitudes of spinning sideband crosspeaks in 2D MAS exchange spectra, in the limit of complete magnetization exchange between the labeled sites, is presented in detail. A new technique that enhances the sensitivity of 2D MAS exchange spectra to molecular structure, called orientationally weighted 2D MAS exchange spectroscopy, is introduced. Symmetry principles that underlie the construction of pulse sequences for orientationally weighted 2D MAS exchange spectroscopy are explained. Experimental demonstrations of the utility of 2D MAS exchange spectroscopy in structural investigations of peptide and protein backbone conformations are carried out on a model C-13-labeled tripeptide, L-alanylglycylglycine. The dihedral angles phi and psi that characterize the peptide backbone conformation at Gly-2 are obtained accurately from the orientationally weighted and unweighted 2D C-13 NMR exchange spectra.