In this thesis, we investigate compact left and right multipliers on and prove that the existence of a non-zero compact left or right multipliers on is equivalent to compactness G. Also, we study the modulus of a right (left) multipliers T., and show that in general the modulus of T need not be a right (left) multiplier. Finally, we investigate compact left and right multipliers on the second dual of M(G) and we give a describtion of compact left multipliers on the second dual of M(G) in term of left completely continuous elements of the second dual of M(G).