Mean-field theory is one of the most standard tools used by
physicists to analyze phase transitions in realistic systems. However,
regarding rigorous proofs, the link to mean-field theory has been
limited to asymptotic statements which do not yield enough control
of the actual systems. In this talk I will describe a new approach to
this set of problems -- developed jointly with Lincoln Chayes and
Nicolas Crawford -- that overcomes this hurdle in a rather elegant
way. As a conclusion, I will show that a general, ferromagnetic
nearest neighbor spin system on Z^d undergoes a first order phase
transition whenever the mean-field theory indicates one, provided
the dimension d is sufficiently large. Extensions to systems with non
nearest neighbor interactions will also be discussed.