Attention is the core mechanism of today's most used architectures for natural language processing and has been analyzed from many perspectives, including its effectiveness for machine translation-related tasks. Among these studies, attention resulted to be a useful source of information to get insights about word alignment also when the input text is substituted with audio segments, as in the case of the speech translation (ST) task. In this paper, we propose AlignAtt, a novel policy for simultaneous ST (SimulST) that exploits the attention information to generate source-target alignments that guide the model during inference. Through experiments on the 8 language pairs of MuST-C v1.0, we show that AlignAtt outperforms previous state-of-the-art SimulST policies applied to offline-trained models with gains in terms of BLEU of 2 points and latency reductions ranging from 0.5s to 0.8s across the 8 languages.