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Image-text matching is gaining a leading role among tasks involving the joint understanding of vision and language. In literature, this task is often used as a pre-training objective to forge architectures able to jointly deal with images and texts. Nonetheless, it has a direct downstream application: cross-modal retrieval, which consists in finding images related to a given query text or vice-versa. Solving this task is of critical importance in cross-modal search engines. Many recent methods proposed effective solutions to the image-text matching problem, mostly using recent large vision-language (VL) Transformer networks. However, these models are often computationally expensive, especially at inference time. This prevents their adoption in large-scale cross-modal retrieval scenarios, where results should be provided to the user almost instantaneously. In this paper, we propose to fill in the gap between effectiveness and efficiency by proposing an ALign And DIstill Network (ALADIN). ALADIN first produces high-effective scores by aligning at fine-grained level images and texts. Then, it learns a shared embedding space - where an efficient kNN search can be performed - by distilling the relevance scores obtained from the fine-grained alignments. We obtained remarkable results on MS-COCO, showing that our method can compete with state-of-the-art VL Transformers while being almost 90 times faster. The code for reproducing our results is available at https://github.com/mesnico/ALADIN.