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Over the past forty years, a substantial body of work has appeared centered around the syzygies of algebraic varieties. Classical results about defining equations have emerged as the first cases of a much more general picture involving higher syzygies. Moreover, as computer-assisted computations have become practical, Castelnuovo-Mumford regularity has come into focus as a measure of algebraic complexity. This research has touched on a wide array of topics in algebraic geometry, and the time seemed ripe for a survey of some of these ideas. The present monograph attempts to provide this. Conceived as an introduction to the theory rather than a comprehensive survey, the authors focus on the geometric side of the story. A first course in algebraic geometry and some exposure to commutative algebra are sufficient background for most of the material, although facility with coherent cohomology is assumed. The presentation is pitched at the level of an intermediate or advanced graduate course.