The work described in this dissertation concerns the development of new methods for performing computer simulations of real materials from first principles or ab initio i.e. using the fundamental equations of quantum mechanics and only well-controlled approximations. In particular, these methods have been developed within the framework of density-functional theory and therefore lie in the realms of both quantum chemistry and computational condensed matter physics. The work is particularly concerned with methods which are efficient in the sense that the computational effort required scales only linearly with system-size (i.e. the volume or number of electrons) whereas traditional methods have scaled with the cube of the system-size which has restricted their range of applicability. The aim of this work is therefore to extend the scope of ab initio quantum-mechanical calculations beyond what is currently possible. Density-functional theory has traditionally been applied by making use of a mapping from the system of interacting electrons to a fictitious system of non-interacting particles. However, the need to maintain the mutual orthogonality of the wave functions of the fictitious system leads to the cubic scaling mentioned above, and is ultimately responsible for limiting the maximum size of systems which can be treated. Making use of a reformulation of the problem in terms of the single-particle density-matrix eliminates the need to work with the wave functions directly. Moreover, exploiting the short-ranged nature of the density-matrix leads in principle to a linear-scaling method. The dissertation tackles two issues which are relevant to obtaining practical schemes for performing linear-scaling calculations. Firstly a localised basis set is developed which is used to describe the density-matrix computationally. Analytic results for several key quantities required by the calculation are derived, namely the overlap, kinetic energy and non-local pseudopotential matrix elements. These results allow accurate calculation of the total energy of the system and have been implemented and tested computationally. Secondly, the dissertation discusses several methods for imposing the difficult non-linear idempotency constraint on the density-matrix.