The Ising model has exact solutions in 1D and 2D. These do not
give the same results as mean-field theory for the critical exponents
and this observation opened
the way for the modern study of phase transitions.
In 3D there are only numerical approaches such as
simulations and series expansions. I will not discuss
the algorithms of these here (thats for the Computational Physics course)
but I will show some important features.
Both give the same numbers up to many figures of precision
so I guess we are getting it right.