  Partial synchronous states appear between full synchrony and asynchrony and
exhibit many interesting properties. Most frequently, these states are studied
within the framework of phase approximation. The latter is used ubiquitously to
analyze coupled oscillatory systems. Typically, the phase dynamics description
is obtained in the weak coupling limit, i.e., in the first-order in the
coupling strength. The extension beyond the first-order represents an unsolved
problem and is an active area of research. In this paper, three partially
synchronous states are investigated and presented in order of increasing
complexity. First, the usage of the phase response curve for the description of
macroscopic oscillators is analyzed. To achieve this, the response of the
mean-field oscillations in a model of all-to-all coupled limit-cycle
oscillators to pulse stimulation is measured. The next part treats a two-group
Kuramoto model, where the interaction of one attractive and one repulsive group
results in an interesting solitary state, situated between full synchrony and
self-consistent partial synchrony. In the last part, the phase dynamics of a
relatively simple system of three Stuart-Landau oscillators are extended beyond
the weak coupling limit. The resulting model contains triplet terms in the
high-order phase approximation, though the structural connections are only
pairwise. Finally, the scaling of the new terms with the coupling is analyzed.
