  Perturbative analyses of planetary resonances commonly predict singularities
and/or divergences of resonance widths at very low and very high
eccentricities. We have recently re-examined the nature of these divergences
using non-perturbative numerical analyses, making use of Poincar\'e sections
but from a different perspective relative to previous implementations of this
method. This perspective reveals fine structure of resonances which otherwise
remains hidden in conventional approaches, including analytical,
semi-analytical and numerical-averaging approaches based on the critical
resonant angle. At low eccentricity, first order resonances do not have
diverging widths but have two asymmetric branches leading away from the nominal
resonance location. A sequence of structures called ``low-eccentricity resonant
bridges" connecting neighboring resonances is revealed. At planet-grazing
eccentricity, the true resonance width is non-divergent. At higher
eccentricities, the new results reveal hitherto unknown resonant structures and
show that these parameter regions have a loss of some -- though not necessarily
entire -- resonance libration zones to chaos. The chaos at high eccentricities
was previously attributed to the overlap of neighboring resonances. The new
results reveal the additional role of bifurcations and co-existence of
phase-shifted resonance zones at higher eccentricities. By employing a
geometric point of view, we relate the high eccentricity phase space structures
and their transitions to the shapes of resonant orbits in the rotating frame.
We outline some directions for future research to advance understanding of the
dynamics of mean motion resonances.
