The Vortex filament flow (VF) is a well known evolution of filaments that when written in terms of the curvature of the filament becomes the well-known non-linear Schrodinger (NLS) equation, a completely integrable PDE. Thus VF is said to be an Euclidean realization of NLS. During the last few decades there has been a flurry of literature discovering new local realizations of most integrable PDEs and investigating the consequences of this and similar connections between geometric curve flows and integrable systems, not only in Euclidean but in most other classical geometries (projective, conformal, symplectic, etc). The studies included the generation of Hamiltonian structures from the background geometry and the lifts of these structures to the curves. In this talk we will briefly describe this connection and will focus on the more recent discrete case, including the effective use of discrete moving frames. For discrete integrable flows and maps most problems remain open.