This is an introduction to a very active field of research, on the boundary between mathematics and physics. It is aimed at graduate students and researchers in geometry and string theory. Proofs or sketches are given for many important results. From the reviews: "An excellent introduction to current research in the geometry of Calabi-Yau manifolds, hyper-Kähler manifolds, exceptional holonomy and mirror symmetry....This is an excellent and useful book." --MATHEMATICAL REVIEWS From the reviews: "Summer schools in Nordfjordeid, Norway, have been organized regularly since 1996. Topics vary each year but there are always three series of lectures by invited experts with evening exercises. […] The themes of all three contributions are interrelated and together they give a nice introduction into a very interesting field of research on the border between mathematics and physics. l would like to strongly recommend the book to anybody interested in the topic." (jbu) European Mathematical Society Newsletter, Sept. 2004, p. 44 "[...] This book is an excellent introduction to current research in the geometry of Calabi-Yau manifolds, hyper-Kähler manifolds, exceptional holonomy and mirror symmetry. [...] This is an excellent and useful book. The different chapters mostly fit together reasonably well [...]. It will be popular with anyone trying to learn about Kähler manifolds with c1=0 and mirror symmetry." Richard P. Thomas, Mathematical Reviews, Clippings from Issue 2004c "Summer schools in Nordfjordeid, Norway, have been organized regularly since 1996. … The school held in June 2001 was devoted to recent interaction between differential and algebraic geometry. The book consists of notes written by lecturers of the corresponding three series of lectures. … they give a nice introduction into a very interesting field of research on the border between mathematics and physics. I would like to strongly recommend the book to anybody interested in the topic." (EMS, September, 2004) "This book is an excellent introduction to current research in the geometry of Calabi-Yau manifolds, hyper-Kähler manifolds, exceptional holonomy and mirror symmetry. … the choice of topics is a sensible one. … This is an excellent and useful book. … It will be popular with anyone trying to learn about Kähler manifolds with c 1 = 0 and mirror symmetry." (Richard P. Thomas, MathematicalReviews, 2004 c)