Analysis of Teukolsky equations on slowly rotating Kerr spacetimes In this thesis we treat the extreme Newman Penrose components of both the Maxwell field (s=±1) and the linearized gravitational perturbations (or linearized gravity for short) (s=±2) in the exterior of a slowly rotating Kerr black hole. Upon different rescalings we can obtain spin s components which satisfy the separable Teukolsky master equation (TME). For each of these spin s components defined in Kinnersley tetrad the resulting equations by performing some first order differential operator on it once and twice (twice only for s=±2) together with the TME are in the form of an inhomogeneous spin weighted wave equation (ISWWE) with different potentials and constitute a linear spin weighted wave system. We then prove energy and integrated local energy decay (Morawetz) estimates for this type of ISWWE and utilize them to achieve both a uniform bound of a positive definite energy and a Morawetz estimate for the regular extreme Newman Penrose components defined in the regular Hawking Hartle tetrad. We also present some brief discussions on mode stability for TME for the case of real frequencies. This says that in a fixed subextremal Kerr spacetime there is no nontrivial separated mode solutions to TME which are purely ingoing at horizon and purely outgoing at infinity. This yields a representation formula for solutions to inhomogeneous Teukolsky equations and will play a crucial role in generalizing the above energy and Morawetz estimates results to the full subextremal Kerr case.