Recent developments in the field of differential geometry have been so extensive that a new book with particular emphasis on current work in Riemannian geometry is clearly necessary. This new text brilliantly serves that purpose and includes an elementary account of twistor spaces that will interest both applied mathematicians and physicists. It presents recent developments in the theory of harmonic spaces, commutative spaces, and mean-value theorems previously only available in the source literature. The final chapter provides the only account available in book form of manifolds known as `Willmore surfaces', illustrated by a series of computer-generated pictures. This book is sure to be welcomed by researchers, teachers, and students interested in the latest developments in differential geometry.