Juha Heinonen (1960-2007) was Professor of Mathematics at the University of Michigan. His principal areas of research interest included quasiconformal mappings, nonlinear potential theory, and analysis on metric spaces. He was the author of over 60 research articles, including several posthumously, and two textbooks. A member of the Finnish Academy of Science and Letters, Heinonen received the Excellence in Research Award from the University of Michigan in 1997 and gave an invited lecture at the International Congress of Mathematicians in Beijing in 2002. Pekka Koskela is Professor of Mathematics at the University of Jyväskylä, Finland. He works in Sobolev mappings and in the associated nonlinear analysis, and he has authored over 140 publications. He gave invited lectures at the European Congress of Mathematics in Barcelona in 2000 and at the International Congress of Mathematicians in Hyderabad in 2010. Koskela is a member of the Finnish Academy of Science and Letters. He received the Väisälä Award in 2001 and the Magnus Ehrnrooth Foundation prize in 2012. Nageswari Shanmugalingam is Professor of Mathematics at the University of Cincinnati. Her research interests include analysis in metric measure spaces, potential theory, functions of bounded variation and quasiminimal surfaces in metric setting. The foundational part of the structure of Sobolev spaces in metric setting was developed by her in her PhD thesis in 1999, and she has also contributed to the development of potential theory in metric setting. Her research contributions were recognized by the College of Arts and Sciences at the University of Cincinnati with a McMicken Dean's Award in 2008. Jeremy T. Tyson is Professor of Mathematics at the University of Illinois, Urbana-Champaign, working in analysis in metric spaces, geometric function theory and sub-Riemannian geometry. He has authored over 40 research articles and co-authored two other books. Tyson has received awards for teaching from the University of Illinois at both the departmental and college level. He is a Fellow of the American Mathematical Society.

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• Preface; 1. Introduction; 2. Review of basic functional analysis; 3. Lebesgue theory of Banach space-valued functions; 4. Lipschitz functions and embeddings; 5. Path integrals and modulus; 6. Upper gradients; 7. Sobolev spaces; 8. Poincaré inequalities; 9. Consequences of Poincaré inequalities; 10. Other definitions of Sobolev-type spaces; 11. Gromov–Hausdorff convergence and Poincaré inequalities; 12. Self-improvement of Poincaré inequalities; 13. An Introduction to Cheeger's differentiation theory; 14. Examples, applications and further research directions; References; Notation index; Subject index.