Abstract. Let M g,A[n] be the moduli stack parametrizing weighted stable curves, and let M g,A[n] be its coarse moduli space. These spaces have been introduced by B. Hassett, as compactifications of Mg,n and Mg,n respectively, by assigning rational weights A = (a 1 , ..., an), 0 < a i 1 to the markings. In particular, the classical Deligne-Mumford compactification arises for a 1 = ... = an = 1. In genus zero some of these spaces appear as intermediate steps of the blow-up construction of M 0,n developed by M. Kapranov, while in higher genus they may be related to the LMMP on M g,n. We compute the automorphism groups of most of the Hassett's spaces appearing in the Kapranov's blow-up construction. Furthermore, if g 1 we compute the automorphism groups of all Hassett's spaces. In particular, we prove that if g 1 and 2g − 2 + n 3 then the automorphism groups of both M g,A [n] and M g,A[n] are isomorphic to a subgroup of Sn whose elements are permutations preserving the weight data in a suitable sense.