Contributions to historical climate change have varied substantially among nations. These differences reflect underlying inequalities in wealth and development, and pose a fundamental challenge to the implementation of a globally equitable climate mitigation strategy. This Letter presents a new way to quantify historical inequalities among nations using carbon and climate debts, defined as the amount by which national climate contributions have exceeded a hypothetical equal per-capita share over time. Considering only national CO2 emissions from fossil fuel combustion, accumulated carbon debts across all nations from 1990 to 2013 total 250 billion tonnes of CO2, representing 40% of cumulative world emissions since 1990. Expanding this to reflect the temperature response to a range of emissions, historical climate debts accrued between 1990 and 2010 total 0.11 °C, close to a third of observed warming over that period. Large fractions of this debt are carried by industrialized countries, but also by countries with high levels of deforestation and agriculture. These calculations could contribute to discussions of climate responsibility by providing a tangible way to quantify historical inequalities, which could then inform the funding of mitigation, adaptation and the costs of loss and damages in those countries that have contributed less to historical warming.