: In this thesis, we study compact and weakly compact multipliers on certain Banach algebras related to locally compact groups G. We frist characteriese compact and weakly compact multipliers on the group algebras and the measure algebra of G and their second dual; we show that the existence of a compact and weakly compact multipliers on such algebras is equivalent to compactness of G. Finally, we investigate Riesz multipliers and multipliers commutating with multipliers of some group algebras.
