McInroy, JustinShpectorov, Sergey2024-04-102024-04-102024-12-12McInroy, J., & Shpectorov, S. (2024). Axial algebras of Jordan and Monster type. In [C. M. Campbell, M. R. Quick, E. F. Robertson, C. M. Roney-Dougal, & D. I. Stewart (Eds.), Groups St Andrews 2022 in Newcastle (pp. 246-294). Cambridge University Press.9781009563222http://hdl.handle.net/10034/628593This material has been accepted for publication by Cambridge University Press, and a revised form will be published in [Groups St Andrews 2022 in Newcastle] edited by [C. M. Campbell, M. R. Quick, E. F. Robertson, C. M. Roney-Dougal, D. I. Stewart]. This version is free to view and download for private research and study only. Not for re-distribution or re-use. © Cambridge University Press & Assessment 2025.Axial algebras are a class of non-associative commutative algebras whose properties are defined in terms of a fusion law. When this fusion law is graded, the algebra has a naturally associated group of automorphisms and thus axial algebras are inherently related to group theory. Examples include most Jordan algebras and the Griess algebra for the Monster sporadic simple group. In this survey, we introduce axial algebras, discuss their structural properties and then concentrate on two specific classes: algebras of Jordan and Monster type, which are rich in examples related to simple groups.https://creativecommons.org/licenses/by-nc-nd/4.0/Axial algebrasJordan algebrasConference Contribution