The Euclid satellite, a medium-class mission by the European Space Agency, is set for launch in 2019 as part of the Cosmic Vision 2015–2025 program. Its primary goal is to understand the accelerated expansion of the universe by measuring the shapes and redshifts of galaxies and the distribution of galaxy clusters over a large fraction of the sky. Euclid will explore the expansion history of the universe and the evolution of cosmic structures, providing insights into dark energy, dark matter, and fundamental physics. The Euclid Theory Working Group has compiled this review to provide a guide to the scientific themes that will underlie the group's activity during the preparation of the Euclid mission. The review focuses on cosmology and fundamental physics, with a strong emphasis on science beyond the current standard models. It discusses five broad topics: dark energy and modified gravity, dark matter, initial conditions, basic assumptions, and questions of methodology in data analysis. The review covers a wide range of topics, including dark energy models, modified gravity theories, dark matter properties, initial conditions, and the testing of basic cosmological hypotheses. It also discusses statistical methods for performance forecasts and the implications of Euclid's data for understanding the universe's evolution. Euclid's mission will provide a detailed reconstruction of the clustering of galaxies out to a redshift of 2 and the pattern of light distortion from weak lensing to a redshift of 3. The two main probes, redshift clustering and weak lensing, are complemented by additional cosmological probes such as cross correlations between the cosmic microwave background and large-scale structure, luminosity distance through supernovae Ia, and the abundance and properties of galaxy clusters. The review also addresses the challenges of testing the basic cosmological assumptions and the potential for new discoveries in the field of cosmology. It highlights the importance of understanding the nature of dark energy and dark matter, as well as the initial conditions of the universe, in order to develop a more complete understanding of the cosmos. The review concludes with a discussion of the statistical methods used to forecast the performance of probes like Euclid and the potential for future developments in the field.The Euclid satellite, a medium-class mission by the European Space Agency, is set for launch in 2019 as part of the Cosmic Vision 2015–2025 program. Its primary goal is to understand the accelerated expansion of the universe by measuring the shapes and redshifts of galaxies and the distribution of galaxy clusters over a large fraction of the sky. Euclid will explore the expansion history of the universe and the evolution of cosmic structures, providing insights into dark energy, dark matter, and fundamental physics. The Euclid Theory Working Group has compiled this review to provide a guide to the scientific themes that will underlie the group's activity during the preparation of the Euclid mission. The review focuses on cosmology and fundamental physics, with a strong emphasis on science beyond the current standard models. It discusses five broad topics: dark energy and modified gravity, dark matter, initial conditions, basic assumptions, and questions of methodology in data analysis. The review covers a wide range of topics, including dark energy models, modified gravity theories, dark matter properties, initial conditions, and the testing of basic cosmological hypotheses. It also discusses statistical methods for performance forecasts and the implications of Euclid's data for understanding the universe's evolution. Euclid's mission will provide a detailed reconstruction of the clustering of galaxies out to a redshift of 2 and the pattern of light distortion from weak lensing to a redshift of 3. The two main probes, redshift clustering and weak lensing, are complemented by additional cosmological probes such as cross correlations between the cosmic microwave background and large-scale structure, luminosity distance through supernovae Ia, and the abundance and properties of galaxy clusters. The review also addresses the challenges of testing the basic cosmological assumptions and the potential for new discoveries in the field of cosmology. It highlights the importance of understanding the nature of dark energy and dark matter, as well as the initial conditions of the universe, in order to develop a more complete understanding of the cosmos. The review concludes with a discussion of the statistical methods used to forecast the performance of probes like Euclid and the potential for future developments in the field.