Deep learning (DL) has proved successful in medical imaging and, in the wake of the recent COVID-19 pandemic, some works have started to investigate DL-based solutions for the assisted diagnosis of lung diseases. While existing works focus on CT scans, this paper studies the application of DL techniques for the analysis of lung ultrasonography (LUS) images. Specifically, we present a novel fully-annotated dataset of LUS images collected from several Italian hospitals, with labels indicating the degree of disease severity at a frame-level, video-level, and pixel-level (segmentation masks). Leveraging these data, we introduce several deep models that address relevant tasks for the automatic analysis of LUS images. In particular, we present a novel deep network, derived from Spatial Transformer Networks, which simultaneously predicts the disease severity score associated to a input frame and provides localization of pathological artefacts in a weakly-supervised way. Furthermore, we introduce a new method based on uninorms for effective frame score aggregation at a video-level. Finally, we benchmark state of the art deep models for estimating pixel-level segmentations of COVID-19 imaging biomarkers. Experiments on the proposed dataset demonstrate satisfactory results on all the considered tasks, paving the way to future research on DL for the assisted diagnosis of COVID-19 from LUS data.