This report is the response to a request by the Committee on Space Research of the International Council for Science to prepare a roadmap on observation and integrated Earth-system science for the coming ten years. Its focus is on the combined use of observations and modelling to address the functioning, predictability and projected evolution of interacting components of the Earth system on timescales out to a century or so. It discusses how observations support integrated Earth-system science and its applications, and identifies planned enhancements to the contributing observing systems and other requirements for observations and their processing. All types of observation are considered, but emphasis is placed on those made from space. The origins and development of the integrated view of the Earth system are outlined, noting the interactions between the main components that lead to requirements for integrated science and modelling, and for the observations that guide and support them. What constitutes an Earth-system model is discussed. Summaries are given of key cycles within the Earth system. The nature of Earth observation and the arrangements for international coordination essential for effective operation of global observing systems are introduced. Instances are given of present types of observation, what is already on the roadmap for 2016-2025 and some of the issues to be faced. Observations that are organised on a systematic basis and observations that are made for process understanding and model development, or other research or demonstration purposes, are covered. Specific accounts are given for many of the variables of the Earth system. The current status and prospects for Earth-system modelling are summarized. The evolution towards applying Earth-system models for environmental monitoring and prediction as well as for climate simulation and projection is outlined. General aspects of the improvement of models, whether through refining the representations of processes that are already incorporated or through adding new processes or components, are discussed. Some important elements of Earth-system models are considered more fully. Data assimilation is discussed not only because it uses observations and models to generate datasets for monitoring the Earth system and for initiating and evaluating predictions, in particular through reanalysis, but also because of the feedback it provides on the quality of both the observations and the models employed. Inverse methods for surface-flux or model-parameter estimation are also covered. Reviews are given of the way observations and the processed datasets based on them are used for evaluating models, and of the combined use of observations and models for monitoring and interpreting the behaviour of the Earth system and for predicting and projecting its future. A set of concluding discussions covers general developmental needs, requirements for continuity of space-based observing systems, further long-term requirements for observations and other data, technological advances and data challenges, and the importance of enhanced international co-operation.