We will report some results concerning the Yamabe problem and the Nirenberg problem. Related topics will also be discussed. Such studies have led to new results on some conformally invariant fully nonlinear equations arising from geometry. We will also present these results which include some Liouville type theorems, Harnack type inequalities, existence and compactness of solutions to some nonlinear version of the Yamabe problem.2000 Mathematics Subject Classification: 35, 58.

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Following the approach in our earlier paper [2] and using the gradient estimates developed in [2] and [3], we give another Liouville type theorem for some conformally invariant fully nonlinear equations. Various Liouville type theorems for conformally invariant equations have been obtained by Obata, Gidas-Ni-Nirenberg, CaffarelliGidas-Spruck, Viaclovsky, Chang-Gursky-Yang, and Li-Li. For these, as well as for related works, see [2] and the references therein.For n ≥ 3, let S n×n be the set of n × n real symmetric matrices, S n×n +

We present some results in [9], a continuation of our earlier works [7] and [8]. One result is the existence and compactness of solutions to a fully nonlinear version of the Yamabe problem on locally conformally flat Riemannian manifolds, and the other is a Harnack type inequality for general conformally invariant fully nonlinear second order elliptic equations.Let (M, g) be an n−dimensional, compact, smooth Riemannian manifold without boundary, n ≥ 3, consider the Weyl-Schouten tensor A g = 1 n−2

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