First we will review the old idea (from J.W. Gibbs and L.Boltzmann circa 1900) of a Gibbs ensemble in statistical physics. Secondly we will survey the modern "rigorous" (i.e. mathematically precisely stateable and, hopefully, provable) theory of Gibbs measures in the statistical physics of classical lattice models. I may briefly mention some new results and one minor open question from my UNT dissertation, if time permits. Thirdly, we will describe how, in the 1960s, the Gibbs measure took on a new life in the emerging field of dynamical systems. This will include a brief survey of the ``thermodynamic formalism" of Gibbs measures in well studied classes of dynamical systems. Finally we will introduce a new type of entropy (joint with M.Urbanski) that opens up new classes of dynamical systems to study by thermodynamic formalism.

The talk will make use of some measure theory and topology and is intended for a generally mathematically literate audience. No prior knowledge of physics or dynamical systems is assumed and all specialized terminology will be defined and briefly explained as it arises.