Over the last sixty years, the probabilistic analysis of models in statistical mechanics has become a dynamic branch of research, with numerous interactions with other fields of mathematics: graph theory, complex analysis, geometric group theory, topology, computer science, to name a few. Initially introduced as a model for porosity, percolation has become fundamental in the theoretical understanding of phase transitions. Developing percolation theory has been a source of several fascinating mathematical challenges, but also has provided a solid mathematical support to key concepts in statistical physics: sharpness of phase transition, renormalization theory, existence of scaling limits and critical exponents, relationship between discrete and continuous descriptions (constructive field theory), universality at criticality, existence of a critical dimension for mean-field behavior... Percolation theory is now mature and well established, but the picture is still not complete: a few basic questions have not found an answer and are now considered as some of the most challenging questions in probability. In particular, two major open problems witness the current lack of understanding in the field: the continuity of the phase transition in dimension three, and the universality in dimension two. In this this project, we aim at developing new tools in order to make progress towards the above mentioned open problems.