The two talks in the workshop give a survey of the course I am planning to give. It is based on joint work with Matthias Aschenbrenner and Joris van der Hoeven.

Since 2017 there has been a viable model theory for the differential field of transseries. More recently we have used this to throw new light on Hardy fields as well, proving for example that maximal Hardy fields all have the same theory as the differential field of transseries, and that any two maximal Hardy fields are back-and-forth equivalent. This amounts to an essentially complete extension theory for Hardy fields. As byproducts we can answer some questions asked by Boshernitzan about Hardy fields. There remain very attractive open problems, and these will be discussed as well.