We cover some background materials in full-nonlinear elliptic PDE, then selected problems in conformal geometry. Topics include: a brief review of some key facts of 2nd order linear and semi-linear elliptic PDE and basic theory of fully non-linear 2nd order elliptic PDE; Garding theory of fully non-linear PDE; brief survey of Yamabe problem in conformal geometry; elementary symmetric functions on tensors; the curvature on compact closed 4-manifolds; works of S. Chen and J. Case-Y. Wang on classes of fully non-linear PDE with matching (fully non-linear) boundary conditions on manifold settings, etc.