Characterizing graphs by the spectra of various matrices associated with the graphs has long been an important topic in spectral graph theory. However, it is generally very hard to show a given graph to be determined by its spectrum. This paper gives a simple criterion for a tournament to be determined by its adjacency spectrum. More precisely, let G be a tournament of order n with adjacency matrix A, and WA(G)=[e,Ae,...,An−1e] be its walk matrix, where e is the all-one vector. We show that, for any self-converse tournament G, if detWA(G) is square-free, then G is uniquely determined by its adjacency spectrum among all tournaments. The result is somewhat unexpected, which was achieved by employing some recent tools developed by the second author for showing a graph to be determined by its generalized spectrum. The key observation is that for a tournament G, the complement of G coincides with the converse of G, and hence a certain equivalence between the ordinary adjacency spectrum and the generalized (skew)-adjacency spectrum for G can be established. Moreover, some equivalent formulations of the above result, as well as some related ones are also presented.