An explicit understanding of the (category of all smooth, complex) representations of p-adic groups provides an important tool within representation theory and beyond with applications in particular to the Langlands program. In this talk, I will likely survey what we know about the construction of all so called supercuspidal representations, which are the building blocks for all representations. For more than 20 years it remained open to extend a general construction to the case p=2, and I will sketch what makes this case so special and how we could overcome the obstacles in my joint work with David Schwein from January 2025.

If time allows, I will also sketch how two preprints from August 2024 with Jeffrey Adler, Manish Mishra and Kazuma Ohara allow us to reduce a lot of problems about the (category of) representations of p-adic groups to problems about representations of finite groups of Lie type, where answers are often already known or are at least easier to achieve.