In the first part of the talk I will discuss the problem of existence of measures of maximal Hausdorff dimension on repellers and (uniformly) hyperbolic sets. In the conformal case, a measure of maximal Hausdorff dimension always exists and is an invariant Gibbs measure corresponding to a specially chosen geometric potential. In general, however, there may be no invariant measures of maximal Hausdorff dimension. I then introduce the notions of Caratheodory measure and associated Caratheodory dimension which are generated by a given Holder continuous potential. This Caratheodory measure can be used as a new tool to develop thermodynamic formalism on locally maximal uniformly hyperbolic sets. In particular, applying this approach to the geometric potential gives a Gibbs measure which is the unique invariant measure of maximal Caratheodory dimension.