Ising model and more complex interactions between particles

[alboot]

Hello,
First of all, sorry if my question is trivial but I'm not a physicist ;-)
I heard a lot about the Ising model used to compute the energy of a system with 2 states. This model seems to be a bit "simple" because it assumes that particles interact with their nearest neighbors only. I was wondering if they were more complicated models that would consider more complicated interactions ?

 

[alboot]

$$E = \sum_{ij} J_{ij} S_i S_j$$
Actually, I was wrong cause we can really put whatever we want in $$J_{ij}$$, is that right ? If we do so, we can also approximate the solution that minimizes the energy function ?