We present a new compactness theory of Ricci flows. This theory states that any sequence of Ricci flows that is pointed in an appropriate sense, subsequentially converges to a synthetic flow. Under a natural non-collapsing condition, this limiting flow is smooth on the complement of a singular set of parabolic codimension at least 4. We furthermore obtain a stratification of the singular set with optimal dimensional bounds depending on the symmetries of the tangent flows. Our methods also imply the corresponding quantitative stratification result and the expected $L^p$-curvature bounds. \\ \\ As an application we obtain a description of the singularity formation at the first singular time and a long-time characterization of immortal flows, which generalizes the thick-thin decomposition in dimension 3. We also obtain a backwards pseudolocality theorem and discuss several other applications. \\ \\ The schedule of the lecture series will be approximately as follows: \\ 1. Heat Kernel and entropy estimates on Ricci flow backgrounds and related geometric bounds. \\ 2. Continuation of Lecture 1 + Synthetic definition of Ricci flows (metric flows) and basic properties \\ 3. Convergence and compactness theory of metric flows \\ 4. Partial regularity of limits of Ricci flows