Projection algorithms for solving (non-convex) feasibility problems in Euclidiean spaces provide powerful and computationally efficient schemes for a wide variety of applications. We focus on the Method of Alternating Projections (MAP) and the Averaged Alternating Reflection Algorithm (AAR) which are the foundation of the state of the art algorithms in imaging and signal processing, our principle application. In the last several years the regularity requirements for linear convergence of these algorithms have come into sharper relief. We focus on two different approaches dealing with non-convex feasibility. One approach (Bauschke, Luke, Phan, Wang 2012) that uses normal cone techniques achieving optimal convergence rates for MAP. The other approach (H.-Luke 2012) deals with direct/primal techniques achieving sufficient and even necessary conditions for linear convergence of both MAP and AAR, however this strategy does not yield optimal quantitative rates. An adequate description of the relation between these two approaches remains open. Closing this gap requires a good understanding of qualitative and quantitative characterizations of set intersections. An overview over different concepts of regularity (e.g. linear regularity, strong regularity, Friedrichs angle,...) will be given in this talk.