Providing a timely description of the present state of the art of moduli spaces of curves and their geometry, this volume is written in a way which will make it extremely useful both for young people who want to approach this important field, and also for established researchers, who will find references, problems, original expositions, new viewpoints, etc. The book collects the lecture notes of a number of leading algebraic geometers and in particular specialists in the field of moduli spaces of curves and their geometry. This is an important subject in algebraic geometry and complex analysis which has seen spectacular developments in recent decades, with important applications to other parts of mathematics such as birational geometry and enumerative geometry, and to other sciences, including physics. The themes treated are classical but with a constant look to modern developments (see Cascini, Debarre, Farkas, and Sernesi's contributions), and include very new material, such as Bridgeland stability (see Macri's lecture notes) and tropical geometry (see Chan's lecture notes).

Articles in this volume are based on lectures given at three conferences on Geometry at the Frontier, held at the Universidad de la Frontera, Pucon, Chile in 2016, 2017, and 2018. The papers cover recent developments on the theory of algebraic varieties--in particular, of their automorphism groups and moduli spaces. They will be of interest to anyone working in the area, as well as young mathematicians and students interested in complex and algebraic geometry.