Internet of Things (IoT) devices will play an important role in emerging applications, since their sensing, actuation, processing, and wireless communication capabilities stimulate data collection, transmission and decision processes of smart applications. However, new challenges arise from the widespread popularity of IoT devices, including the need for processing more complicated data structures and high dimensional data/signals. The unprecedented volume, heterogeneity, and velocity of IoT data calls for a communication paradigm shift from a search for accuracy or fidelity to semantics extraction and goal accomplishment. In this paper, we provide a partial but insightful overview of recent research efforts in this newly formed area of goal-oriented (GO) and semantic communications, focusing on the problem of GO data compression for IoT applications.