I present numerical studies of perturbed black holes and neutron stars. The behaviour of such compact objects is of particular interest now because the gravitational wave signals which they emit may be the first to be detected by the new generation of interferometric detectors. A single perturbed black hole is well approximated using the linearized Einstein equations and the problem is reduced to solving a simple wave equation with a potential. I show how such a wave equation may be evolved numerically on Cauchy or characteristic hypersurfaces. I present a new numerical code for evolving scalar wave perturbations in Kerr spacetime as a characteristic (null-timelike) initial value problem,. This code suffers from an instability but this problem can be pushed to very late times by increasing the grid resolution. The code shows second order convergence up until the time that the instability takes over. Previous numerical studies of perturbations in Kerr spacetime [2] were carried out using a Cauchy evolution code. I use this code to contribute to a recent discussion over the late-time behaviour of perturbations with initial form m = 0, l = 4, finding the expected fall-off of t⁻³. I also use this code to lend support to the superradiance resonance cavity interpretation of Glampedakis and Andersson [52] to explain the long-lived quasinormal modes of Kerr spacetime. I have adapted the Cauchy code to use coordinates (r_*, θ_*) more suitable for a characteristic evolution and this code gives the expected results, comparable with the original version. I have also developed a code to evolve scalar field perturbations in Kerr spacetime in double-null coordinates but this is not stable. The cosmological constant plays an important role in cosmology and particle physics and will also effect the asymptotic geometry of black hole spacetimes. I investigate the effect of a cosmological constant on the late-time behaviour of a perturbed scalar field in Kerr-de Sitter spacetime and present some new results which reveal apparently undamped oscillations in some cases. I also investigate superradiance in the presence of a cosmological constant. The results show super-radiance in the expected frequency range but the superradiance seems to extend beyond the lower frequency limit predicted by Khanal [70] at large Λ. Evolutions of neutron star spacetimes are more complicated than the corresponding problem for black holes due to the presence of matter. Not only must we consider the response of the exterior spacetime but also the behaviour of the material that makes up the star. There are many more factors which need to be taken into account. I present work done in collaboration with Watts and Andersson [3] to study the effect of differential rotation on a simple system, a rotating spherical shell. For this I have developed a numerical time evolution code which confirms the predictions of Watts et al. [4] for a new class of oscillations and a new instability.