It is well known that Lie groups and homogeneous spaces provide a rich source of interesting examples for a variety of geometric aspects. Likewise it is often the case that topological and geometric restrictions yield the existence of isometries in a more or less direct way. The most obvious example of this is the group of deck transformations of the universal cover of a nonsimply connected manifold. More subtle situations arise in the contexts of rigidity problems and of collapsing with

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... curvature. Our main purpose here is to present the view point that the geometry of isometry groups provide a natural and useful link between theory and examples in Riemannian geometry. This fairly unexplored territory is fascinating and interesting in its own right. At the same time it enters naturally when such groups arise in settings as above. More importantly, perhaps, this study also provides a systematic search for geometrically interesting examples, where the group of isometries is short of acting transitively in contrast to the case of homogeneous spaces mentioned above. Although the general philosophy presented here applies to many different situations, we will illustrate our point of view primarily within the context of manifolds with nonnegative or positive curvature. We have divided our presentation into five sections. The first section is concerned with basic equivariant Riemannian geometry of smooth compact transformation groups, including a treatment of Alexandrov geometry of orbit spaces. Section two is the heart of the subject. It deals with the geometry and topology in the presence of symmetries. It is here we explain our guiding principle which provides a systematic search for new constructions and examples of manifolds of positive or nonnegative curvature. In the third section we exhibit all the known constructions and examples of such manifolds. The topic of section four is geometry via symmetries. We display three different types of problems in which symmetries are not immediately present from the outset, but where their emergence is crucial to their solutions. In the last section we discuss a number of open problems and conjectures related either directly, potentially or at least in spirit to the subject presented here. Our exposition assumes basic knowledge of Riemannian geometry, and a rudimentary familiarity with Lie groups. Although we use Alexandrov geometry of spaces with a lower curvature bound our treatment does not require prior knowledge of this subject. Our intentions have been that anyone with these prerequisites will be able to get an impression of the subject, and guided by the references provided here will be able to go as far as their desires will take them.