Geometric Aspects of the Painlevé Equations $\mathrm{PIII(D_6)}$ and $\mathrm{PIII(D_7)}$

The Riemann–Hilbert approach for the equations $\mathrm{PIII(D_6)}$ and $\mathrm{PIII(D_7)}$ is studied in detail, involving moduli spaces for connections and monodromy data, Okamoto–Painlevé varieties, the Painlevé property, special solutions and explicit Bäcklund transformations.