In this talk, I will report on some recent joint work with Biao Ma and Wei Wei. We will start with some background materials on conformal goemetry. First, we discuss sigma_2 curvature and Q curvature problems for smooth 4-manifolds. Then we define the conic 4-folds. We then summarize our main results. In each case, we obtain Gauss-Bonnet-Chern formulae and discuss geometric implications of our results and possible future works.