Explore the intricate connections between dynamics, algebra, and geometry in free-by-cyclic groups in this 50-minute lecture by Jean Pierre Mutanguha. Delve into the construction of semi-direct products of finitely generated free groups and integers, examining their Cayley graphs and other geometric models. Investigate the dynamics of defining automorphisms, algebraic properties of free-by-cyclic groups, and the coarse geometry of their models. Discover how these concepts are influenced by Thurston's work on surface homeomorphisms and 3-manifolds. Gain insights into various results, from classic to recent, and learn key ideas behind them. Encounter numerous open problems in this field, providing ample opportunities for further research and exploration.