Ammonia is a promising new energy carrier that has the potential to replace traditional fossil fuels due to its high calorific value, CO₂ emission free and renewable properties, and low storage and transportation costs. However, the ammonia combustion faces several challenges such as high NO_x emission, slow combustion rate, and ignition difficulties. To address these challenges, selective catalytic combustion is an efficient and promising way that can decrease combustion temperature, enhance combustion stability, improve fuel conversion rate and reduce NO_x emission, while requiring less auxiliary fuel and consuming less energy. Copper-based catalysts are cost-effective and they are demonstrated to exhibit excellent performance in ammonia catalytic combustion with high N₂ selectivity. In this review, the reaction experimental setup and primary mechanisms, namely NH-HNO, N₂H₄ and i-SCR pathways are presented firstly; then the recent research progress on copper-based supported catalysts for ammonia combustion, including the effect of preparation methods, types of support materials, metal-support interactions, and Cu-based single or multiple active components on catalytic performance are reviewed in detail. Finally, the review is concluded by summarizing the research achievements, current drawbacks and proposing the future developments.