Robot awareness of human actions is an essential research problem in robotics with many important real-world applications, including human-robot collaboration and teaming. Over the past few years, depth sensors have become a standard device widely used by intelligent robots for 3D perception, which can also offer human skeletal data in 3D space. Several methods based on skeletal data were designed to enable robot awareness of human actions with satisfactory accuracy. However, previous methods treated all body parts and features equally important, without the capability to identify discriminative body parts and features. In this paper, we propose a novel simultaneous Feature And Body-part Learning (FABL) approach that simultaneously identifies discriminative body parts and features, and efficiently integrates all available information together to enable real-time robot awareness of human behaviors. We formulate FABL as a regression-like optimization problem with structured sparsity-inducing norms to model interrelationships of body parts and features. We also develop an optimization algorithm to solve the formulated problem, which possesses a theoretical guarantee to find the optimal solution. To evaluate FABL, three experiments were performed using public benchmark datasets, including the MSR Action3D and CAD-60 datasets, as well as a Baxter robot in practical assistive living applications. Experimental results show that our FABL approach obtains a high recognition accuracy with a processing speed of the order-of-magnitude of 101 Hz, which makes FABL a promising method to enable real-time robot awareness of human behaviors in practical robotics applications.