In the field of biomedical imaging, ultrasonography has become common practice, and used as an important auxiliary diagnostic tool with unique advantages, such as being non-ionizing and often portable. This article reviews the state-of-the-art in medical ultrasound (US) image processing and in particular its applications in the examination of the lungs. First, we briefly introduce the basis of lung US (LUS) examination. We focus on (i) the characteristics of lung ultrasonography and (ii) its ability to detect a variety of diseases through the identification of various artifacts exhibiting on LUS images. We group medical US image computing methods into two categories: 1) model-based methods and 2) data-driven methods. We particularly discuss inverse problem-based methods exploited in US image despeckling, deconvolution, and line artifacts detection for the former, while we exemplify various works based on deep/machine learning (ML), which exploit various network architectures through supervised, weakly supervised, and unsupervised learning for the data-driven approaches.