Most liquids gradually solidify at low temperature via a physical process called the glass transition towards a non-equilibrium disordered state of matter. This process is well-known experimentally at the macroscopic scale. At the fundamental level, however, the statistical mechanics description of the phenomenon is much less advanced, as it took several decades of difficult analytic work to `only' derive a solid mean-field transition of the liquid-glass transition. Considerable progress was also made to develop simple yet realistic atomistic models for glass transition studies, as well as numerical methods to more efficiently sample the configuration space which is known to be highly complex. Demonstrating the existence of the transition and studying the associated properties (universality, exponents, characteristic lengthscales) has not been possible so far. We wish to solve this difficult problem. In this thesis, we will develop and combine numerical approaches to systematically investigate the statistical mechanics nature of the transition between liquid and glass states in equilibrium conditions. Ultimately, this work will provide a definitive answer to a mystery that has haunted the field of disordered systems for more than fifty years by demonstrating whether a glass state of matter can truly be defined in three-dimensional glass-forming liquids.