Abstract: Let G be a connected finite graph. Backman, Baker, and Yuen have constructed a family of explicit and easy-to-describe bijections $g_{\sigma,\sigma^*}$ between spanning trees of G and $(\sigma,\sigma^*)$-compatible orientations, where the $(\sigma,\sigma^*)$-compatible orientations are the representatives of equivalence classes of orientations up to cycle-cocycle reversal which are determined by a cycle signature $\sigma$ and a cocycle signature $\sigma^*$. Their proof makes use of zonotopal subdivisions and the bijections $g_{\sigma,\sigma^*}$ are called geometric bijections. Recently we have extended the geometric bijections to subgraph-orientation correspondences. In this talk, I will introduce the bijections and the geometry behind them.