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This comprehensive and self-contained textbook presents an accessible overview of the state of the art of multivariate algorithmics and complexity. Increasingly, multivariate algorithmics is having significant practical impact in many application domains, with even more developments on the horizon. The text describes how the multivariate framework allows an extended dialog with a problem, enabling the reader who masters the complexity issues under discussion to use the positive and negative toolkits in their own research. Features: describes many of the standard algorithmic techniques available for establishing parametric tractability; reviews the classical hardness classes; explores the various limitations and relaxations of the methods; showcases the powerful new lower bound techniques; examines various different algorithmic solutions to the same problems, highlighting the insights to be gained from each approach; demonstrates how complexity methods and ideas have evolved over the past 25 years.