Consider a spacetime which approaches a Schwarzschild solution. We will discuss the following problem: Assuming decay of appropriate norms of the Ricci rotation coefficients and their derivatives, can one prove boundedness/ decay for the curvature components and their derivatives? The talk will give a positive answer to this question and explain some of the difficulties arising from the fact that not all curvature components decay. As an important ingredient, we generalize recent work of Dafermos and Rodnianski regarding decay for the wave equation to the setting of the Bianchi equations.