This is an announcement for the paper “Geometry of $\ell_p^n$-balls: Classical results and recent developments” by Joscha Prochno<https://arxiv.org/search/math?searchtype=author&query=Prochno%2C+J>, Christoph Thäle<https://arxiv.org/search/math?searchtype=author&query=Th%C3%A4le%2C+C>, Nicola Turchi<https://arxiv.org/search/math?searchtype=author&query=Turchi%2C+N>. Abstract: This survey will appear as a chapter in the forthcoming "Proceedings of the HDP VIII Conference". The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1808.10435