Despite significant progress in Video Large Language Models (Video-LLMs) for offline video understanding, existing online Video-LLMs typically struggle to simultaneously process continuous frame-by-frame inputs and determine optimal response timing, often compromising real-time responsiveness and narrative coherence. To address these limitations, we introduce LiveStar, a pioneering live streaming assistant that achieves always-on proactive responses through adaptive streaming decoding. Specifically, LiveStar incorporates: (1) a training strategy enabling incremental video-language alignment for variable-length video streams, preserving temporal consistency across dynamically evolving frame sequences; (2) a response-silence decoding framework that determines optimal proactive response timing via a single forward pass verification; (3) memory-aware acceleration via peak-end memory compression for online inference on 10+ minute videos, combined with streaming key-value cache to achieve 1.53x faster inference. We also construct an OmniStar dataset, a comprehensive dataset for training and benchmarking that encompasses 15 diverse real-world scenarios and 5 evaluation tasks for online video understanding. Extensive experiments across three benchmarks demonstrate LiveStar's state-of-the-art performance, achieving an average 19.5% improvement in semantic correctness with 18.1% reduced timing difference compared to existing online Video-LLMs, while improving FPS by 12.0% across all five OmniStar tasks. Our model and dataset can be accessed at https://github.com/yzy-bupt/LiveStar.