The book provides a state-of-the-art description of the construction and properties of coherent systems on algebraic curves and their moduli spaces, including many results based on the research works of the author and his collaborators. This is a developing theory which generalizes the classical theory of linear systems and has applications to higher rank BrillNoether theory, and projective embeddings of curves and syzygies. Coherent Systems on Algebraic Curves begins by describing the construction of the moduli spaces and their basic properties, before proceeding on to construction methods, coherent systems in genus 0 and 1, existence of coherent systems in higher genus, irreducibility and smoothness of the moduli spaces, and special results for rank 2 and coherent systems on special curves. Many well-worked examples and open problems are included, and links with the projective geometry of curves, emphasized.