chapter one is devoted to a moderate discussion on preliminaries, according to our requirements. chapter two which is based on our work in (24) is devoted introducting weighted semigroups (s, w), and studying some famous function spaces on them, especially the relations between go (s, w) and other function speces are invesigated. in fact this chapter is a complement to (32). one of the main features of this chapter is that the background set is a semigroup and nit a group. in some instances we have even tired to drop some conditions (such as local compactness, or beign hausdoff) from the semigroup. moreover, up to the best of our knowledge, for the first time, we have investigated, the connection between the translation invariance of go (s) and go (s, w), and consequently the connection between the topological structure of s, and the translation invariance of go (s, w) has been investigated. this will be useful in investigating the relation between go (s, w) and other function spaces. chapter three, is devoted to introducing means, homomorphisms and compactifications. and studying the relations between m-admissible subalgebra of c (s, w) and compactifications of (s, w). this chapter shows that the one to one correspondence between m-admissible subalgebras, and compactifications reduces to the inclusion relation, in the case of weighted semigroups, i. e. compactifications lose a great deal of their importance in this case. moreover we show that the existence of compactifications is independent of the definition of amean (for which there has been quite a few different ones). this, in fact, is a consequence of the definition of a homomorphism between two weighted semigroups. for the analytic background of this thesis, we refer to (6), (7), (20), while for topological background we refer to (23).