Explore the fascinating world of finitely presented simple groups of homeomorphisms of the real line in this 39-minute conference talk from the Workshop on "Geometric and Asymptotic Group Theory with Applications 2023 - Groups and Dynamics" at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI). Begin with an introduction to Thompson's groups F and T, then delve into two families of examples of finitely generated simple groups of homeomorphisms of the real line. Discover a third family that is also finitely presented. Gain insights into the joint work of the speaker with Yash Lodha and Cristobal Rivas, expanding your understanding of geometric and asymptotic group theory.