We address the challenges of Byzantine-robust training in asynchronous distributed machine learning systems, aiming to enhance efficiency amid massive parallelization and heterogeneous compute resources. Asynchronous systems, marked by independently operating workers and intermittent updates, uniquely struggle with maintaining integrity against Byzantine failures, which encompass malicious or erroneous actions that disrupt learning. The inherent delays in such settings not only introduce additional bias to the system but also obscure the disruptions caused by Byzantine faults. To tackle these issues, we adapt the Byzantine framework to asynchronous dynamics by introducing a novel weighted robust aggregation framework. This allows for the extension of robust aggregators and a recent meta-aggregator to their weighted versions, mitigating the effects of delayed updates. By further incorporating a recent variance-reduction technique, we achieve an optimal convergence rate for the first time in an asynchronous Byzantine environment. Our methodology is rigorously validated through empirical and theoretical analysis, demonstrating its effectiveness in enhancing fault tolerance and optimizing performance in asynchronous ML systems.