Stability (local surjectivity) and rigidity of the scalar curvature have been studied in an early work of Fischer-Marsden on \vacuum static spaces". Inspired by this line of research, we seek similar properties for Q-curvature by studying \Q-singular spaces", which were introduced by Chang-Gursky-Yang. In this talk, we investigate deformation problems of Q-curvature on closed Rie- mannian manifolds with dimensions n 3. In particular, we classify nonnegative Einstein Q-singular spaces and prove local surjectivity for non-Q-singular spaces. We also prove local rigidity of at manifolds. For global results, we show that any smooth functions can be realized as a Q- curvature on generic Q- at manifolds. However, a locally conformally at metric on n-tori with nonnegative Q-curvature has to be at. This is joint work with Wei Yuan.