Professor Lauret is the recipient of the 2007 Ramanujan Prize for Young Mathematicians from Developing Countries. Abstract: The construction of Einstein Riemannian metrics on manifolds is a classical problem in differential geometry and general relativity. General existence and non-existence results are hard to obtain, and it is a natural simplification to impose additional symmetry assumptions. In this talk, we will first give an elementary introduction to the study of G-orbits and the orbit space V/G, for a subgroup G of the group of invertible linear maps GL(V), where V is a real finite dimensional vector space. Strong results from this theory will then be applied to the study of Einstein manifolds admitting certain transitive groups of symmetries.