I'm interested in models used before the Hubbard model (free electron model, band model, etc, both itinerant and localized models). What were their strengths and weaknesses? Which model is currently the main topic of modern research? Did the Hubbard model overcome any obstacles that other models didn't?
I'm certainly no expert on the history of these models, and can only answer from my working knowledge about the topic. (In particular, I'm not quite the right person to give a detailed comparison with the Hubbard model.) Nevertheless, the text below is too long for a comment, and it might still be of some help. I do not claim completeness, but think that these are the bare minimum one should know about in this context. I'd be interested to see more detailed answers by people who know more about this than I do!
The Hubbard model is from the 1960s. Thus the Ising model (Lenz, 1920) is certainly older, as is the Heisenberg model (Heisenberg, 1928) for (anti)ferromagnetism. See e.g. "Magnetism" at Wikipedia, "Ferromagnetism" at Wikipedia and references there for more.
The Ising model is a classical-physical model, in the sense that the "spins" are discrete, and as such can be thought of as describing the limit of highly anisotropic interactions between the magnetic dipoles of the atoms. The Ising model in 1d was already solved by Ising in his PhD thesis; the 2d Ising model (on a square lattice) was famously tackled by Onsager (1944); while in 3d and up it is still topic of active research -- see e.g. references at "Ising model" at Wikipedia.
The Heisenberg model is already interesting in 1d (spatial), and shows up in current research, ranging from topics such as "quantum integrability", to "quench dynamics", to mathematically rigorous probabilistic approaches.