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To reduce their environmental impact, cloud datacenters' are increasingly focused on optimizing applications' carbon-efficiency, or work done per mass of carbon emitted. To facilitate such optimizations, we present Carbon Containers, a simple system-level facility, which extends prior work on power containers, that automatically regulates applications' carbon emissions in response to variations in both their work-load's intensity and their energy's carbon-intensity. Specifically, Carbon Containers enable applications to specify a maximum carbon emissions rate (in g.CO2e/hr), and then transparently enforce this rate via a combination of vertical scaling, container migration, and suspend/resume while maximizing either energy-efficiency or performance. Carbon Containers are especially useful for applications that i) must continue running even during high-carbon periods, and ii) execute in regions with few variations in carbon-intensity. These low-variability regions also tend to have high average carbon-intensity, which increases the importance of regulating carbon emissions. We implement a Carbon Container prototype by extending Linux Containers to incorporate the mechanisms above and evaluate it using real workload traces and carbon-intensity data from multiple regions. We compare Carbon Containers with prior work that regulates carbon emissions by suspending/resuming applications during high/low carbon periods. We show that Carbon Containers are more carbon-efficient and improve performance while maintaining similar carbon emissions.