In this thesis we study various aspects of the problem of finding rational linear spaces on hypersurfaces. This problem can be approached by the Hardy-Littlewood circle method, establishing a Local-Global Principle provided that the hypersurface is 'sufficiently non-singular' and the number of variables is large enough. However, the special structure of the linear spaces allows us to obtain some improvement over previous approaches. A generalised version is also addressed, which allows us to count linear spaces under somewhat more flexible conditions. We then investigate the local solubility. In particular, by adopting a new approach to the analysis of the density of p-adic solutions arising in applications of the circle method, we show that under modest conditions the existence of non-trivial p-adic solutions suffices to establish positivity of the singular series. This improves on earlier approaches due to Davenport, Schmidt and others, which require the existence of non-singular p-adic solutions. Finally, we exhibit the strength of our methods by deriving unconditional results concerning the existence of linear spaces on systems of cubic and quintic equations.