Our aim is to present several applications of a version of Mourre theory that we have recently developed. We can easily deduce from it, for example, a very precise form of the limiting absorption principle for perturbations $H = h(P) + V_S+V_L$ of a constant coefficient pseudo-differential operator $h(P)$ by short-range and long-range non local potentials $V_S$ and $V_L$. The perturbations $V_S,V_L$ are quite singular locally (the sum above is required to exist only in form-sense) and the assumptions concerning their behaviour at infinity are essentially optimal (e.g. $V_S$ is of Enss type). Furthermore, if such an $H$ is perturbed by another short-range potential, the relative wave operators exist and are complete. The theory works also for systems (like Dirac operators). Other applications are to division theorems, i.e. properties of the operators of multiplication by $(h(x)\pm i_0)^{-1}$, under minimal regularity assumptions on $h$. In particular these examples show that the regularity assumptions we make in our abstract version of Mourre theory are essentially optimal.