In thesis, at frist, with due attention to the relation between approximation property and approximable operators on Banach spaces, we study approximation property, and describe some important results and implicatinos of this property. We also show that each Hilbert space has the approximation property. We also prove that if X is a Banach space, then the collection of approximable operators on X, is a Banach algebra. and we invertigate their relation to compact operators.