The classical Nirenberg problem asks for which functions on the sphere arise as the scalar curvature of a metric that is conformal to the standard metric. The problem is equivalent to solving a second order elliptic equation. In this talk, we will discuss similar questions for some fractional order curvatures. This is equivalent to solving some nonlocal equations of order up to n/2, where n is the dimension. We will show existence and compactness results. We will also discuss related topics such as fractional Yamabe flows and those with isolated singularities.