The central object of the study in this thesis is ΩO(<I>n</I>), the space of closed continuous loops on an orthogonal group O(<I>n</I>) based at the identity-element 1 Ε O(<I>n</I>). The space ΩO(<I>n</I>) carries a group structure given by pointwise multiplication of paths in the group O(<I>n</I>). This makes it an infinite dimensional Lie group. A filtration of ΩO(<I>n</I>), more precisely of the subspace of 'polynomial' loops, is constructed. This can be thought of as the 'real' analogue of the Mitchell-Richter filtration of ΩSU(<I>n</I>). Our filtration of ΩO(<I>n</I>) splits stably and O(<I>n</I>)-equivariantly in the cases <I>n</I> = 3, 4. We obtain: In contrast to the complex case no general splitting result can hold (this follows from work by Hopkins on stable indecomposability of ΩSp(2)). The thesis also investigates the topology of the loopspace of a real Stiefel-manifold. A stable O(<I>n</I>)-equivariant splitting for the fibrewise loop-space of a projective bundle is used to give a splitting for the free loop-space LRP<I>ⁿ</I> on a real projective space.