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Recent years have witnessed an emerging paradigm shift toward embodied artificial intelligence, in which an agent must learn to solve challenging tasks by interacting with its environment. There are several challenges in solving embodied multimodal tasks, including long-horizon planning, vision-and-language grounding, and efficient exploration. We focus on a critical bottleneck, namely the performance of planning and navigation. To tackle this challenge, we propose a Neural SLAM approach that, for the first time, utilizes several modalities for exploration, predicts an affordance-aware semantic map, and plans over it at the same time. This significantly improves exploration efficiency, leads to robust long-horizon planning, and enables effective vision-and-language grounding. With the proposed Affordance-aware Multimodal Neural SLAM (AMSLAM) approach, we obtain more than 40% improvement over prior published work on the ALFRED benchmark and set a new state-of-the-art generalization performance at a success rate of 23.48% on the test unseen scenes.