Summary: We consider solutions \((M^4, g(t))\), \(0 \leq t < T\), to Ricci flow on compact, four-dimensional manifolds without boundary. We prove integral curvature estimates which are valid for any such solution. In the case when the scalar curvature is bounded and \(T < \infty \), we show that these estimates imply that the (spatial) integral of the square of the norm of the Riemannian curvature is bounded by a constant independent of time \(t\) for all \(0 \leq t < T\) and that the space time integral over \(M \times [0, T)\) of the fourth power of the norm of the Ricci curvature is bounded.