This thesis is concerned with the application of Laser Microprobe Mass Spectrometry (LAMMS) to the microanalysis of inorganic materials and in particular to the quantification of such analyses. The investigation consists both of an assessment of the capabilities of a LAMMS instrument, the Cambridge Mass Spectrometry LIMA 2A, and an attempt to correlate experimental results with theoretical predictions. The principles of the operation of a LAMMS instrument are discussed and a description of the LIMA 2A instrument is presented. A survey of the literature concerned with the interaction of a high-powered laser with a solid specimen to produce a plasma, the basis of the LAMMS technique, is included. Particular emphasis is given to the description of this interaction in terms of a model based on a local thermodynamic equilibrium (LTE) in the plasma. This model is applied to the conditions estimated to be produced in the LIMA instrument to make simple predictions of expected results. A discussion of the possible methods for converting LAMMS data into a quantitative analysis is given, along with a brief description of the statistical techniques used for data handling. Several model materials, principally chosen for their well-defined composition and known lateral homogeneity, are used in this work. These are single-crystal silicon, a binary copper-nickel alloy and three III-V semiconductors drawn from the Ga-In-As system. The effect of the instrument itself is investigated, with a view to establishing its contribution to the errors observed in the data. This is followed by an investigation of the variation of both the absolute and relative ion signals produced. The variations in the relative ion signals are then compared with the predictions of the LTE model in an attempt to establish its validity. This comparison is also used to estimate the conditions produced in the laser-induced plasma and their variation with specimen chemistry and laser power density. The general conclusions of the investigation are drawn together in a discussion of the preferred methods for the quantification of LAMMS data and the expected error in the resulting analysis. It is shown that, provided appropriate methods are used, the LAMMS technique can provide quantitative analyses with precisions of about 5-20%.