The one-dimensional Hubbard model, as for its higher dimensional counterparts, is a remarkable model containing the essence of the physics of correlated quantum systems in one dimension. This model can be approached by methods ranging from exact (Bethe ansatz) solutions, field theory (Bosonization) and numerical methods (such as DMRG). I will show how the combination of these three lines of attack allows one to get an essentially complete description of the physics, and how this physics differs (or not) from the physics of higher dimensional systems. I will also discuss how such physics can and has been observed experimentally both in condensed matter but also with cold atomic gases.
Finally, time permitting, I will address some of the open challenges, in particular in going from one to higher dimensions.
