This is the first account in book form of the theory of harmonic morphisms between Riemannian manifolds. Harmonic morphisms are maps which preserve Laplace's equation. They can be characterized as harmonic maps which satisfy an additional first order condition. Examples include harmonic functions, conformal mappings in the plane, and holomorphic functions with values in a Riemann surface. There are connections with many concepts in differential geometry, for example, Killing fields, geodesics, foliations, Clifford systems, twistor spaces, Hermitian structures, isoparametric mappings, and Einstein metrics, and also the Brownian path-preserving maps of probability theory. Giving a complete account of the fundamental aspects of the subject, this book is self-contained, assuming only a basic knowledge of differential geometry. One chapter follows the complete development of the fundamental geometric aspects of harmonic maps from scratch. This text is suitable for a beginning graduate student interested in harmonic maps and morphisms, or related subjects. The student is brought to the frontiers of knowledge in this rapidly expanding field in which there are many interesting avenues of research to be developed. The authors are world leaders in the field, and have established many of the key results. In this book they have brought together their work and the work of many others to form a coherent account of the subject. This book is the 29th volume in the London Mathematical Society Monographs series, published by Oxford University press on behalf of the London Mathematical Society. The series contains authoritative accounts of current research in mathematics and high quality expository works bringing the reader to the frontiers of research. Of particular interest are topics that have developed rapidly in the past ten years or so, but which have reached a certain level of maturity. Clarity of exposition is important and each book contains preliminary material to make the topic accessible to those commencing work in this area.