Geometric Analysis Learning Seminar:  We consider k-Hessian operators(and when k=n, it is the Monge Amp\’{e}re operator) as convex envelopes of linear operators. And then we will do the fractional analogue of k-Hessian operators, and show the existence and semiconcavity of the solutions of the fractional k-Hessian equations, under a relatively simple framework of global solutions prescribing data at infinity and global barriers. In addition, the uniformly ellipticity of the fractional k-Hessian operators can be proved, thus the regularity results of nonlocal equations developed by L.Caffarelli and L.Silvestre can be applied to prove C^{2s+\alpha} regularity.