The theory of o-minimal structures provides a powerful framework for the study of geometrically tame structures. In the past couple of decades, a deep link connecting o-minimality to structures arising from algebraic and arithmetic geometry has been developing. It has been clear, however, that the axioms of o-minimality do not fully capture some algebro-arithmetic aspects of tameness that one expcts to find in these naturally-arising structures. We propose a notion of "sharply o-minimal" structures extending the standard axioms of o-minimality in an attempt to incorporate these refined aspects into the theory.

I will outline through conjectures and various partial results the potential development of the "sharp" theory in parallel to the standard one. I will also illustrate some applications of this emerging theory in diophantine geometry, focusing on the role it plays in the (very recent) proof of the Andre-Oort conjecture for general Shimura varieties, and on polynomial-time computability questions for some diophantine problems.