Past warm periods provide an opportunity to evaluate climate models under extreme forcing scenarios, in particular high (> 800 ppmv) atmospheric CO2 concentrations. Although a post-hoc intercomparison of Eocene (~50 million years ago, 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 latest Paleocene and the early Eocene. Together these form the first phase of DeepMIP – the deeptime model intercomparison project, itself a group within the wider Paleoclimate Modelling Intercomparison Project (PMIP). The experimental design consists of three core paleo simulations and a set of optional sensitivity studies. The experimental design specifies and provides guidance on boundary conditions associated with palaeogeography, greenhouse gases, orbital configuration, solar constant, land surface parameters, and aerosols. Initial conditions, simulation length, and output variables are also specified. Finally, we explain how the geological datasets, which will be used to evaluate the simulations, will be developed.