Although non-commutative geometry is historically the latest tool introduced to understand topological phases in quantum matter — such as the integer quantum Hall effect — it is one of the most successful one in terms of its range of applications, including for disordered systems. In this talk I will briefly set the problem of the classification of quantum phases and explain why NCG fits this problem naturally in a non-interacting approximation. I will then present recent results that go beyond this approximation, in particular a cohomological index associated with loops of quantum states.