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This thesis consists of three discrete parts. The first is concerned with the study of the ferromagnetic phase transition in the Ising spin andfrac12; model, with random substitutional impurities. Series in the impurity concentration for the susceptibility are extended by one or two terms for the cubic, square and face-centred cubic lattices, using a set of rules which cuts the number of clusters to be considered. Closed analytic form for all terms in the series are obtained. An eight term series for the tetragonal zircon lattice is also developed. Phase boundaries and critical indices are obtained by use of ratio methods and Padandeacute; approximants. Comparison of the results, particularly for the critical index, with recent high temperature series work is made, and the single pole assumption usually made for the divergence of the susceptibility is critically examined. High temperature series for the dilute Ising model with a transverse field are also generated for the tetragonal zircon (7 terms) and face-centred cubic lattices (6 terms): the results for the former are compared with recent experimental work, and some general conclusions are drawn from the latter series about the shape of the phase surface in concentration, temperature and transverse field. Some features of the behaviour of the critical index are also examined. In the second part the description of the two-magnon Raman scattering profile in dilute antiferromagnets is attempted by calculation of the first few moments. Using the Nandeacute;el state as an approximation, the zero^th, first, second and third moments are calculated at zero temperature: the zero^th and second moments are also evaluated in the infinite temperature limit. As the profile at low temperatures is thought to consist of two peaks, various attempts are made to construct moments that will describe solely the two-magnon peak. Reasonable agreement with experiment at low and high temperatures is found, and some comparison with low temperature experiments on dilute MnF₂ is made. Temperature and concentration dependent forms for the zero^th, first and second moments are also derived, using the constant coupling approximation to provide a consistent treatment of thermal averages of one and two spins, and comparison with experiment on pure NiF₂ is made. The third part is concerned with the location of the phase transition in more random systems: a CPA-like treatment for the Fourier transformed spin correlation function is developed and applied to the cubic Ising spin andfrac12; model with next neighbour interactions randomly removed, with some degree of success. A more random model due to Handrich is also considered, and the transition temperature is found to rise as fluctuations in the interaction increase, if the average interaction is held constant. Comparison with a constant coupling theory due to Handrich which predicts the reverse is made.

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