A. Chang, J. Qing, and P. Yang proved a conformal gap theorem for Bach-flat metrics with a round sphere as the model case more than ten years ago. In this talk, we extend this result to prove
conformally invariant gap theorems for Bach-flat 4-manifolds with Fubini-Study metric on complex projective space and product metric on the product of spheres as model cases. An iteration argument plays an important role in the case of complex projective space and the convergence theory of Bach-flat metrics is the main analytical tool in the case of the product of spheres. If time permits, I will discuss some recent progress.