Deep networks have brought significant advances in robot perception, enabling to improve the capabilities of robots in several visual tasks, ranging from object detection and recognition to pose estimation, semantic scene segmentation and many others. Still, most approaches typically address visual tasks in isolation, resulting in overspecialized models which achieve strong performances in specific applications but work poorly in other (often related) tasks. This is clearly sub-optimal for a robot which is often required to perform simultaneously multiple visual recognition tasks in order to properly act and interact with the environment. This problem is exacerbated by the limited computational and memory resources typically available onboard to a robotic platform. The problem of learning flexible models which can handle multiple tasks in a lightweight manner has recently gained attention in the computer vision community and benchmarks supporting this research have been proposed. In this work we study this problem in the robot vision context, proposing a new benchmark, the RGB-D Triathlon, and evaluating state of the art algorithms in this novel challenging scenario. We also define a new evaluation protocol, better suited to the robot vision setting. Results shed light on the strengths and weaknesses of existing approaches and on open issues, suggesting directions for future research.