We revisit the well-known Curve Shortening Flow for immersed curves in the d-dimensional Euclidean space. We exploit a fundamental structure of the problem to derive a new global construction of a solution, that is, a construction that is valid for all times and is insensitive to singularities. The construction is characterized by discretization in time and the approximant, while still exhibiting the possible formation of finitely many singularities at a finite set of singular times, exists globally. It is also well behaved and simpler to analyze than a solution of the CSF, which is obtained in the limit. Estimates for a natural (geometric) norm involving length and total absolute curvature allow passage to the limit. Many classical qualitative results about the flow can be recovered by exploiting the simplicity of the approximant and new ones can be proved. The construction also suggests a numerical procedure for the computation of the flow which proves very effective as demonstrated by a series of numerical experiments shown at the end of the talk.