This is the final version of the article. Available from European Geosciences Union via the DOI in this record. ; Past warm periods provide an opportunity to evaluate climate models under extreme forcing scenarios, in particular high (> 800ppmv) atmospheric CO2 concentrations. Although a post hoc intercomparison of Eocene (∼ 50 Ma) climate model simulations and geological data has been carried out previously, models of past high-CO2 periods have never been evaluated in a consistent framework. Here, we present an experimental design for climate model simulations of three warm periods within the early Eocene and the latest Paleocene (the EECO, PETM, and pre-PETM). Together with the CMIP6 pre-industrial control and abrupt 4 × CO2 simulations, and additional sensitivity studies, these form the first phase of DeepMIP-the Deep-time Model Intercomparison Project, itself a group within the wider Paleoclimate Modelling Intercomparison Project (PMIP). The experimental design specifies and provides guidance on boundary conditions associated with palaeogeography, greenhouse gases, astronomical configuration, solar constant, land surface processes, and aerosols. Initial conditions, simulation length, and output variables are also specified. Finally, we explain how the geological data sets, which will be used to evaluate the simulations, will be developed. ; We thank NERC grant NE/N006828/1 for providing funds for the first DeepMIP meeting in Boulder, Colorado, USA, in January 2016. Daniel J. Lunt acknowledges the NERC grant “Cretaceous Paleocene–Eocene: Exploring Climate and Climate Sensitivity” (NE/K014757/1), and advanced ERC grant “The Greenhouse Earth System” (T-GRES, project reference 340923), awarded to Rich Pancost. Matthew Huber acknowledges funding from NSF OCE-0902882. Michiel L. J. Baatsen, Henk A. Dijkstra, and Anna S. von der Heydt acknowledge support by the Netherlands Earth System Science Centre (NESSC), financially supported by the Ministry of Education, Culture and Science (OCW), 024.002.001. We thank two anonymous reviewers whose comments were very useful in improving and clarifying the experimental design.