This thesis presents (a) a survey of the use of the conformal group from its beginnings to the present time, and (b) a determination of those algebraically special vacuum Einstein space-times with an expanding and/or twisting congruence of null geodesics, which locally possess a homothetic symmetry as well as a Killing symmetry (isometry). Unless the space-time is Petrov type N with twist-free geodesic rays, one can restrict attention to one proper homothetic motion plus the assumed Killing motion(s). The formalism developed to undertake the systematic search for such vacuum space-times is an extension of the tetrad formalism used by Debney,Kerr & Schild G.C. Debney, R.P. Kerr & A. Schild, J.Math.Phys. 10, 1842 (1969). and by Kerr & Debney R.P. Kerr & G.C. Debney, J.Math.Phys. 11, 2807 (1970). The spaces which admit one homothetic Killing vector (HKV) plus 2,3 or 4 Killing vectors (KVs) are completely determined. There are 9 such metrics (12 with 3 degeneracies) - one admitting 4 KVs, one with 3 KVs, and seven with 2 KVs. Those spaces which admit one HKV plus one KV are not completely determined owing to the field equations not being solved in some cases. However, 9 metrics are found, many of which appear to be new. Petrov type N vacuum spaces with expansion and/or twist which admit a homothety are possible when one KV of special type is also present, or when the homothety alone is of special type. An extensive bibliography is given.