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The development of atomic layer epitaxial growth techniques for semiconductors has allowed structures to be fabricated in which layers of different composition (and therefore different bandgap) are successively deposited. Layer thicknesses are typically a few tens of nanometres and may even be of unit cell dimensions. The characterisation of these materials is not only interesting in its own right, but essential to the correct interpretation of their electronic and optoelectronic properties. A variety of analysis techniques can be applied, but transmission electron microscopy (TEM) is one of the few capable of probing these structures over relatively large areas, whilst yielding results which are free from lateral averages over distances which are large by comparison with the layer thickness. The work described was performed with the Al<SUB>x</SUB>Ga<SUB>1-x</SUB>As system in mind, but the results are also relevant to the characterisation of other semiconductor alloys. Conventionally, chemical analysis is performed in the TEM using either energy dispersive spectroscopy or electron energy loss spectroscopy. However both of these lack the necessary sensitivity and spatial resolution for the analysis of Al<SUB>x</SUB>Ga<SUB>1-x</SUB>As. This thesis therefore describes the development and application of characterisation methods for semiconductor heterostructures which make use of the TEM imaging and diffraction properties of the material. Firstly, the general requirements which must be met by a characterisation method or methods are described, and the currently available techniques are reviewed. The potential use of convergent beam diffraction is then considered, and it is demonstrated that, whilst this technique might be applicable to the analysis of ternary alloys such as Ga<SUB>1-x</SUB>In<SUB>x</SUB>As and GaSb<SUB>y</SUB>As<SUB>1-y</SUB>, even dynamical effects in patterns are rarely sensitive to the composition of Al<SUB>x</SUB>Ga<SUB>1-x</SUB>As in any quantitatively useful way. Composition measurement using 002 dark field intensities is shown to be a more sensitive and accurate technique, although questions are raised about the adequacy of accepted theories of image contrast for the treatment of inelastic scattering effects in this particular case. Other techniques are also considered briefly, including Fresnel defocus contrast analysis, and the use of a trace analysis method for the determination of heterostructure layer orientations and layer thicknesses. Finally, the application of these techniques in practice is demonstrated, taking as examples the characterisation of 'graded layer' structures, and an investigation of the effects of Se<SUP>+</SUP> implantation on a GaAs/Al<SUB>x</SUB>Ga<SUB>1-x</SUB>As superlattice.

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