chapters 1 and 2 establish the basic theory of amenability of topological groups and amenability of banach algebras. also we prove that. if g is a topological group, then r (wluc (g)) (resp. r (luc (g))) if and only if there exists a mean m on wluc (g) (resp. luc (g)) such that for every wluc (g) (resp. every luc (g)) and every element d of a dense subset d od g, m (r)m (f) holds. chapter 3 investigates relations between amenability of banach algebras and groups (semigroups). we show that a a-algebra a is amenable if u (a), the unitary group of a, is amenable. furthermore, give an example that the converse is not true in general. also we prove that. if g is a bounded subset of a unital banach algebra a such that g is a group w. r. t. multiplication operation of a, and span (g)a. if g is a topological group w. r. t. o (a, a)-topology and g is amenable, then a is an amenable banach algebra. thus, we show that: the following statements are equivalent for a von neumann algebra m, with unitary group h and isometry semigroup i. (1) m is injective. (2) there exists a right invariant mean on wluc (h). (3) there exists a right invariant mean on wluc (i). also, the following statements are equivalent for a g-algebra a, with unitary group gg and isometry semigroup s. (1) a is nuclear. (2) there exists a righ invariant mean on luc (g). (3) there exists a right invariant mean on luc (s).