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This paper surveys the current state in analyzing and tuning of subdivision algorithms. These two aspects of subdivision algorithms are very much intertwined with the differential geometry of the subdivision surface. This paper deals with the interconnection of these different aspects of subdivision algorithms and surfaces.The principal idea for the analysis of a subdivision algorithm dates back to the late 70s although the overall technique is only well understood since the early 90s. Most subdivision algorithms are analyzed today but the proofs involve time consuming computations. Only recently simple proofs for a certain class of subdivision algorithms were developed that are based on geometric reasoning. This allows for easier smoothness proofs for new developed or tuned subdivision algorithms.The analysis of the classical algorithms, such Catmull-Clark, Loop, etc., shows that the subdivision surfaces at the extraordinary points are not as smooth as the rest of the surface. It was also shown that the subdivision surfaces of these classical algorithms cannot model certain basic shapes. One way to tune a stationary subdivision algorithms to overcome this problem is to drop the stationarity while at the same time using the smoothness proof of the stationary algorithms.