Capturing the `mutual gaze' of people is essential for understanding and interpreting the social interactions between them. To this end, this paper addresses the problem of detecting people Looking At Each Other (LAEO) in video sequences. For this purpose, we propose LAEO-Net, a new deep CNN for determining LAEO in videos. In contrast to previous works, LAEO-Net takes spatio-temporal tracks as input and reasons about the whole track. It consists of three branches, one for each character's tracked head and one for their relative position. Moreover, we introduce two new LAEO datasets: UCO-LAEO and AVA-LAEO. A thorough experimental evaluation demonstrates the ability of LAEONet to successfully determine if two people are LAEO and the temporal window where it happens. Our model achieves state-of-the-art results on the existing TVHID-LAEO video dataset, significantly outperforming previous approaches. Finally, we apply LAEO-Net to social network analysis, where we automatically infer the social relationship between pairs of people based on the frequency and duration that they LAEO.