This paper strives for the detection of real-world anomalies such as burglaries and assaults in surveillance videos. Although anomalies are generally local, as they happen in a limited portion of the frame, none of the previous works on the subject has ever studied the contribution of locality. In this work, we explore the impact of considering spatiotemporal tubes instead of whole-frame video segments. For this purpose, we enrich existing surveillance videos with spatial and temporal annotations: it is the first dataset for anomaly detection with bounding box supervision in both its train and test set. Our experiments show that a network trained with spatiotemporal tubes performs better than its analogous model trained with whole-frame videos. In addition, we discover that the locality is robust to different kinds of errors in the tube extraction phase at test time. Finally, we demonstrate that our network can provide spatiotemporal proposals for unseen surveillance videos leveraging only video-level labels. By doing, we enlarge our spatiotemporal anomaly dataset without the need for further human labeling.