We review recent studies of spin dynamics in rare-earth orthorhombic perovskite oxides of the type $RM$O$_3$, where $R$ is a rare-earth ion and $M$ is a transition-metal ion, using single-crystal inelastic neutron scattering (INS). After a short introduction to the magnetic INS technique in general, the results of INS experiments on both transition-metal and rare-earth subsystems for four selected compounds (YbFeO$_3$, TmFeO$_3$, YFeO$_3$, YbAlO$_3$) are presented. We show that the spectrum of magnetic excitations consists of two types of collective modes that are well separated in energy: gapped magnons with a typical bandwidth of $<$70 meV, associated with the antiferromagnetically (AFM) ordered transition-metal subsystem, and AFM fluctuations of $<$5 meV within the rare-earth subsystem, with no hybridization of those modes. We discuss the high-energy conventional magnon excitations of the 3$d$ subsystem only briefly, and focus in more detail on the spectacular dynamics of the rare-earth sublattice in these materials. We observe that the nature of the ground state and the low-energy excitation strongly depends on the identity of the rare-earth ion. In the case of non-Kramers ions, the low-symmetry crystal field completely eliminates the degeneracy of the multiplet state, creating a rich magnetic field-temperature phase diagram. In the case of Kramers ions, the resulting ground state is at least a doublet, which can be viewed as an effective quantum spin-1/2. Equally important is the fact that in Yb-based materials the nearest-neighbor exchange interaction dominates in one direction, despite the three-dimensional nature of the orthoperovskite crystal structure. The observation of a fractional spinon continuum and quantum criticality in YbAlO$_3$ demonstrates that Kramers rare-earth based magnets can provide realizations of various aspects of quantum low-dimensional physics.
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