Delays frequently occur in real-world environments, yet standard reinforcement learning (RL) algorithms often assume instantaneous perception of the environment. We study random sensor delays in POMDPs, where observations may arrive out-of-sequence, a setting that has not been previously addressed in RL. We analyze the structure of such delays and demonstrate that naive approaches, such as stacking past observations, are insufficient for reliable performance. To address this, we propose a model-based filtering process that sequentially updates the belief state based on an incoming stream of observations. We then introduce a simple delay-aware framework that incorporates this idea into model-based RL, enabling agents to effectively handle random delays. Applying this framework to the Dreamer world-modeling scheme, our method consistently outperforms delay-aware baselines developed for MDPs and demonstrates robustness to delay distribution shifts during deployment. Additionally, we present experiments on simulated robotic tasks, comparing our method to common practical heuristics and emphasizing the importance of explicitly modeling observation delays.