DescriptionWe classify Mobius invariant differential operators of second order in two dimensional Euclidean space, and study the existence and compactness of the sigma_2-Nirenberg problem on S^2. Furthermore, towards establishing those results, we also obtained Liouville type theorems of solutions, local derivatives estimates, and Bocher type theorems, which are crucial to study the blow-up phenomena of the solutions of sigma_2 equations on S^2.