The coadjoint orbits of a Lie group are naturally symplectic homogeneous manifolds. For compact groups such orbits 'resemble' flag manifolds. The structure of such orbits is reasonably well understood. On the other hand, non-compact and non-semisimple Lie groups (such as the Euclidean or Poincare groups) no longer have forthcoming descriptions of the coadjoint orbits owing to the adjoint and coadjoint representations being inequivalent. In this talk I aim to describe these orbits as 'affine flag manifolds' and highlight an intriguing bijection between adjoint and coadjoint orbits which preserves homotopy type.