Extensions and limits of gravity in three dimensions

General Relativity (GR) has resisted the test of time even a hundred years after its conception. However, GR loses precision and predictability not only at strong gravity regimes but also there are some unexplained phenomena in the weak gravity regimes. There have been a lot of efforts in studying possible deformations of GR because of the experimental and theoretical importance. It is technically very challenging to have an extension of GR, because the new theory would have to deal with the large scale problems taking into account at the same time the behaviour that will arise in the quantum theory. In other words, we are searching for a mathematically consistent theory, capable of reproducing GR at cosmological scales, that can provide an explanation of today's acceleration of the universe but at the same time, capable of giving a description of gravity following the principles of quantum mechanics. Since this task has proved to be very complicated, the aim of this work is to set only a first step to dissect GR and to study different extensions and limits of the theory in order to have a better understanding of it. In this thesis we therefore address the question: What are the possible ways of modifying and studying different limits of gravity? In this thesis we studied two extensions: adding new symmetries (supersymmetry) and adding new parameters (a mass parameter), and two limits: the non-relativistic limit and the ultra-relativistic limit