The European Union (EU) recently declared its intention to implement a carbon border adjustment mechanism (CBAM) to address carbon leakage and competitiveness concerns associated with elevated climate ambitions in its Green Deal. Current literature points to uncertainties regarding the CBAM's effectiveness and compatibility with international trade rules. This study numerically evaluates how alternative EU CBAM designs under various international reactions affect global and regional GHG emissions, outputs and trade flows. Our modeling results confirm substantial carbon leakages and output reductions in the EU's emissions-intensive trade-exposed (EITE) sectors when implementing the Green Deal. We find that the design of the CBAM matters: while a “non-discriminatory” CBAM based on the EU's own scope 1 emission intensities fails to effectively reduce leakages, an ‘aggressive’ CBAM based on exporters’ scope 1 & 2 emission intensities can achieve such goal (albeit with potentially high implementation cost). International retaliations by non-EU countries can only partially offset the EU's gains from the ‘aggressive’ CBAM, while their cooperation can result in smaller losses to the EU EITE sectors and lower leakages. Finally, the CBAM cannot change within-EU imbalances in the EITE sectors, as several EU members lose EITE outputs consistently in all scenarios with and without retaliations.