Riemannian Geometry includes results discovered during the last few years, which have previously only been available through research papers. It contains an elementary account of twistor spaces, of interest to applied mathematicians and physicists, and the final chapter gives the only account available in book form of Willmore surfaces (illustrated by a series of computer-generated pictures). This book is intended for pure mathematicians; applied mathematicians and physicists (especially those interested in twistor methods and applications of differential geometry). Suitable for 2nd and 3rd year undergraduates, and postgraduates - could be used as a 3rd year undergraduate text for a course in Riemannian Geometry.