We report a detailed study of single-ion anisotropy and crystal-field effects in rare-earth cuprates R2CuO4 (R=Nd, Pr, and Sm). It is found that most of the magnetic properties are mainly due to the coupling between the copper and rare-earth magnetic subsystem which exhibits a large single-ion anisotropy. This anisotropy prefers ordering of rare-earth moments along [100] for R=Pr and Nd and along [001] for R=Sm. Combined with a pseudodipolar interaction arising from the anisotropy of the R-Cu exchange, we can explain the magnetic structures of these materials. The spin reorientation transitions in Nd2CuO4 can be explained in terms of a competition between various interplanar interactions which arises because of the rapid temperature dependence of the Nd moment below about 100 K. Finally we introduce a simple two-dimensional model for the Nd spin-wave spectrum. For zero wave vector, this model gives two optical modes involving Cu spins whose temperature-dependent energies agree with experimental results and an acoustic mode whose energy is predicted to be of order √(2k4Δ)≈5μeV, where k4 is the fourfold in-plane anisotropy constant and Δ is the Nd doublet splitting.