In these notes, we present a geometric point of view on diffeomorphic and optimal transport flows, driven by applications in shape analysis, computational anatomy, and machine learning. This perspective brings tools that facilitate the design, the derivation and the study of geodesic and gradient flows on diffeomorphisms and measures. Along with this geometric perspective, we provide a rigorous treatment of the analysis required to study these flows, particularly the geodesic flows on diffeomorphism groups. The final part studies gradient flows arising in machine learning through the lenses of these geometric and analytic perspectives.