Geometric Aspects of the Painlev\'e Equations $\mathrm{P}\mathrm{I}\mathrm{I}\mathrm{I}({\mathrm{D}}_6)$ and $\mathrm{P}\mathrm{I}\mathrm{I}\mathrm{I}({\mathrm{D}}_7)$

The Riemann-Hilbert approach for the equations $\mathrm{P}\mathrm{I}\mathrm{I}\mathrm{I}({\mathrm{D}}_6)$ and ${\rm PIII(D_7)}$ is studied in detail, involving moduli spaces for connections and monodromy data, Okamoto-Painlev'e varieties, the Painlev'e property, special solutions and explicit B"acklund transformations.