This book introduces key topics on Geometric Invariant Theory, a technique to obtaining quotients in algebraic geometry with a good set of properties, through various examples. It starts from the classical Hilbert classification of binary forms, advancing to the construction of the moduli space of semistable holomorphic vector bundles, and to Hitchins theory on Higgs bundles. The relationship between the notion of stability between algebraic, differential and symplectic geometry settings is also covered.Unstable objects in moduli problems -- a result of the construction of moduli spaces -- get specific attention in this work. The notion of the Harder-Narasimhan filtration as a tool to handle them, and its relationship with GIT quotients, provide instigating new calculations in several problems. Applications include a survey of research results on correspondences between Harder-Narasimhan filtrations with the GIT picture and stratifications of the moduli space of Higgs bundles.Graduate students and researchers who want to approach Geometric Invariant Theory in moduli constructions can greatly benefit from this reading,  whose key prerequisites are general courses on algebraic geometry and differential geometry.

Alfonso Zamora Saiz is a Professor at the School of Computer Science Engineering, Technical University of Madrid, Spain. Holding a PhD in algebraic geometry from the Complutense University of Madrid (2013), he has been a visiting PhD student at Cambridge University and Columbia University, a postdoc at the IST in Lisbon, Lecturer at the California State University Channel Islands and a Professor at the CEU San Pablo University in Madrid. His research interests include algebra, geometry and topology in pure mathematics, as well as data analytical applications and mathematics education.Ronald A. Zúñiga-Rojas is a Professor at the School of Mathematics, University of Costa Rica (UCR), and is currently a member of both Center of Mathematical and Meta-Mathematical Research (CIMM-UCR) and the Center of Pure and Applied Mathematics Research (CIMPA-UCR). He completed the Doctors Degree in Mathematics at University of Porto, Portugal, in 2015, in a PhD Programin association with the University of Coimbra in Portugal. His research interests lay on pure mathematics, focused on algebraic geometry, algebraic topology, and differential geometry.

Preface.- Introduction.- Preliminaries.- Geometric Invariant Theory.- Moduli Space of Vector Bundles.- Unstability Correspondence.- Stratifications on the Moduli Space of Higgs Bundles.- References.- Index.