The Euclid Satellite, a European Space Agency mission, aims to understand the origin of the universe's accelerated expansion. It will explore the expansion history and evolution of cosmic structures by measuring galaxy shapes, redshifts, and the distribution of galaxy clusters over a large fraction of the sky. The primary focus is on dark energy and modified gravity, but the science covers a broad range, including cosmology, galaxy evolution, and planetary research. This review, prepared by the Euclid Theory Working Group, covers five main topics: dark energy and modified gravity, dark matter, initial conditions, basic assumptions, and data analysis methodology. It discusses various models of dark energy and modified gravity, the properties of dark matter, the impact of initial conditions on cosmological observations, challenges to the Copernican Principle, and statistical methods for performance forecasts. The review aims to guide the scientific themes that will underlie the group's activities during the mission preparation.The Euclid Satellite, a European Space Agency mission, aims to understand the origin of the universe's accelerated expansion. It will explore the expansion history and evolution of cosmic structures by measuring galaxy shapes, redshifts, and the distribution of galaxy clusters over a large fraction of the sky. The primary focus is on dark energy and modified gravity, but the science covers a broad range, including cosmology, galaxy evolution, and planetary research. This review, prepared by the Euclid Theory Working Group, covers five main topics: dark energy and modified gravity, dark matter, initial conditions, basic assumptions, and data analysis methodology. It discusses various models of dark energy and modified gravity, the properties of dark matter, the impact of initial conditions on cosmological observations, challenges to the Copernican Principle, and statistical methods for performance forecasts. The review aims to guide the scientific themes that will underlie the group's activities during the mission preparation.