This paper presents a novel abstract argumentation framework, called Multi-Attack Argumentation Framework (MAAF), which supports different types of attacks. The introduction of types gives rise to a new family of non-standard semantics which can support applications that classical approaches cannot, while also allowing classical semantics as a special case. The main novelty of the proposed semantics is the discrimination among two different roles that attacks play, namely an attack as a generator of conflicts, and an attack as a means to defend an argument. These two roles have traditionally been considered together in the argumentation literature. Allowing some attack types to serve one of those roles only, gives rise to the different semantics presented here.