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While large language models (LLMs) have substantially expanded support for non-English languages, their ability to generalize across typologically diverse languages and produce contextually appropriate, culturally grounded responses remains a significant challenge. This tutorial examines the rapidly evolving area of understanding, evaluating, and improving the multilingual capabilities and multicultural alignment of LLMs. We will primarily focus on cutting-edge advances and latest publications from 2025–2026, while incorporating selected introductory material to provide necessary background and to ensure accessibility for a broad audience. Participants will gain an up-to-date view of both methods and practical insights, including multilingual pre-, mid- and post-training, multilingual-specific optimization techniques, culturally aware evaluation, and key open problems in developing more fair, inclusive, and robust LLMs.