This book introduces a number of ideas in differential geometry centred around recent work on isoparametric submanifolds. After an initial introduction to the geometry of submanifolds, the author discusses the idea of focal points and principal curvatures. The topic is treated from a totally new geometric point of view where the geometry of the set of critical normals is taken as a way of discussing relations between the principal curvatures. Recent work on holonomy group of the normal bundle of a submanifold due to Thorbergsson, Olmos, Heintze and others are described. The examples of isoparametric submanifolds due to Ferus, Karcher and Munzner is considered in some detail.