Abstract: Ubiquitous sensors generate prohibitively large data sets. Large volumes of such data are nowadays generated by a variety of applications such as imaging platforms and mobile devices, surveillance cameras, social networks, power networks, to list a few. In this era of data deluge, it is of paramount importance to gather only the data that is informative for a specific task in order to limit the required sensing cost, as well as the related costs of storing, processing, or communicating the data. The main goal of this talk is therefore to present topics that transform classical sensing methods, often based on Nyquist-rate sampling, to more structured low-cost sparse sensing mechanisms designed for specific inference tasks, such as estimation, filtering, and detection. More specifically, we present fundamental tools to achieve the lowest sensing cost with a guaranteed performance for the task at hand. Applications can be found in the areas of radar, multi-antenna communications, remote sensing, and medical imaging.