We cover selected topics of some recent developments in conformal geometry, including boundary behavior of asymptotic hyperbolic manifolds; and, problems of existence, uniqueness and compactness of conformal filling in of Poincare-Einstein manifolds with some given classes of metrics on the conformal infinity. We also cover earlier works of J. Lee, Graham-Lee, Qing, Dutta-Jaharavi, Li-Qing-Shi and proceed with recent works by Chang-Ge, Chang-Qing-Ge and others on the subject.