In robust control problems it is often convenient to consider maps between controller sets that are defined by Möbius transformations. Moreover, these loop-transformations are also intimately related to different factorizations, that simplify the structure of the problem. Starting from a fundamental observation that relates internal stability of the control loops to mere stability of a specific LFT, the paper provides an exhaustive answer to the question under which conditions the internal stability of a control loop is preserved by performing a loop-transformation. As a main result it is shown that Möbius transformations defined by unimodular matrices preserves the internal stability of the loop and an explicit formula is also given that relates the two loops. This result is formulated in a general context that includes LTV or LPV systems as well. The paper also provides an overview on the different transformation techniques, as Möbius transformations, LFTs, different generalized chain scattering (CSD) transforms, and their interrelations. This knowledge gives a solid basis for those approaches that use an input-output framework in combination with the IQC analysis and synthesis techniques in the solution of linear parameter varying (LPV) design problems.