This book chapter is published open access.
This article is based on the lecture that I had the honor and pleasure to deliver at the 8th European Congress of Mathematics in Portorož, Slovenia (originally planned for June 2020, then shifted to June 2021 for public health reasons). In the talk I tried to give an overview of some issues linked to torsion in algebraic groups, focusing on some recent research. Taking into account the purposes of reaching a large audience of mathematicians, from all subjects, I started with elementary general concepts, recalling some historical steps, before shifting to more specific themes which I was more familiar with. In these notes, I maintained the same principles, and only slightly expanded the contents of the lecture; indeed, I have not gone into any detailed argument.