In this monograph, some new techniques are developed to define derivation on a Beurling algebra. Moreover, the amenability, weak amenability of a Beurling algebra and amenability of its second dual are discussed. Also, it is shown that if G is non-discrete, then the Banach algebra M(G,w) is not weakly amenable. Finally, the conjecture of H.G.Dales and A.T.M.Lau about 2-weakly amenability of a Beurling algebra is proved with an additional condition which is weaker than almost invariance.

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