Developments in this field have been so extensive since the author's work Introduction to Differential Geometry that a new book with particular emphasis on recent work in Riemannian geometry is clearly necessary. This new work includes an elementary account of twistor spaces which will interest both applied mathematicians and physicists as well as presenting recent developments in the theory of harmonic spaces, commutative spaces, and mean-value theorems previously only available in research papers. 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.