The background of this thesis is the cosmic censorship hypothesis, which states that the gravitational collapse of physically reasonable matter should not result in the formation of naked singularities. In the absence of a proof of this hypothesis, much effort has been directed towards examining spacetimes which contain naked singularities, in an attempt to determine the nature of the cosmic censor. One area of particular interest is the study of perturbations in naked singularity spacetimes. Should perturbations of a spacetime diverge on the Cauchy horizon associated with the naked singularity, then this spacetime can be ruled out as a serious counter-example to cosmic censorship. In this thesis we examine the behaviour of general linear perturbations of the class of self-similar Lemaître-Tolman-Bondi spacetimes which contain a naked singularity. The perturbations naturally split into two classes, odd and even parity, which we consider in turn. For the odd parity perturbation, we first identify a single gauge invariant scalar which describes the perturbation and obeys an inhomogeneous wave equation. We then show that a perturbation which evolves from initially regular data remains finite on the Cauchy horizon. Finiteness is demonstrated by considering the behaviour of suitable energy norms of the perturbation (and pointwise values of these quantities) on natural spacelike hypersurfaces. For the even parity perturbations, we first show that a particular average of the state variable describing the perturbations generically diverges at the Cauchy horizon. Using this, we show that the L^p-norm of the perturbations also diverges, for 1 ≤ p ≤ ∞. This divergence has a characteristic form that depends only on the background spacetime. By combining these results with an extension of odd parity methods, as well as some theorems from real analysis, we can demonstrate that the perturbations generically diverge pointwise on the Cauchy horizon. A general perturbation is a sum of odd and even perturbations; our results therefore indicate that a general perturbation diverges on the Cauchy horizon. This result supports the cosmic censorship hypothesis.