Let G be a graph with n vertices, and let A (G) and D (G) denote respectively the adjacency matrix and the degree matrix of G. Define A α (G)= α D (G)+(1− α) A (G) for any real α∈[0, 1]. The collection of eigenvalues of A α (G) together with multiplicities are called the A α-spectrum of G. A graph G is said to be determined by its A α-spectrum if all graphs having the same A α-spectrum as G are isomorphic to G. We first prove that some graphs are determined by their A α-spectra for 0≤ α< 1, including the complete graph K n, the union of cycles, the complement of the union of cycles, the union of copies of K 2 and K 1, the complement of the union of copies of K 2 and K 1, the path P n, and the complement of P n. Setting α= 0 or 1 2, those graphs are determined by A-or Q-spectra. Secondly, when G is regular, we show that G is determined by its A α-spectrum if and only if the join G∨ K m (m≥ 2) is determined by its A α-spectrum for 1 2< α< 1. Furthermore, we also show that the join K m∨ P n (m, n≥ 2) is determined by its A α-spectrum for 1 2< α< 1. In the end, we pose some related open problems for future study.