The problem of constructing Higgs sectors, that is. finding Higgs representations and potentials that will produce a given spontaneous breakdown G → H (or hierarchy G → H → K...) is considered from a general point of view. The concepts of ordering of little groups, of little spaces and of symmetric algebras are shown to be of use, and it is also shown that in many cases a sum-of-squares potential can be very effective. A number of applications are presented and the pseudo-Goldstone problem is discussed
