This thesis investigates methods of assessing students' mathematical ability by using the computer. It starts by reviewing the general types of assessment within mathematics educational software and then describes some different ways of presenting the assessment on the computer by the use of varying types of questions. In Chapter 2 there is a review of the literature and research conducted in the area of computer assessment of mathematics. In particular, the most prevalent dilemmas of computer aided learning and computer aided assessment are highlighted whilst looking forward at how the contents of further chapters in the thesis can help in addressing some of these difficulties. The following chapter gives an historical account of how the CALM(1) software has addressed some of the inherent difficulties of assessment and highlights the ways in which some of these hurdles have been overcome. The shortfalls of CALM are described and, where relevant, pointers to the parts of thesis which tackle these shortfalls are given. In particular, the work in Chapter 4 undertakes an improvement in the way simple mathematical expressions(2) can be handled as it shows how binary tree constructions can be utilised within an educational environment. Chapter 5 tests out two applications of the binary tree structures with the creation of a tool to aid student-computer communication of mathematics and by providing a method of comparing student-set questions against a true answer. The following chapter describes an educational experiment which set out to show how a computer can be used to assess students' mathematical ability during a formal university examination. It deals with very important educational issues which arise when performing such examinations and gives conclusions as to their educational validity. In particular, issues of student input, partial credit, objectivity, consistency, flexibility and efficiency are considered along with the impact that this research could have for future testing of mathematics. The final chapter describes how the thesis has been instrumental in further research and development within the field of computer assessment of mathematics. (1)CALM is the acronym for the Computer Aided Learning in Mathematics project at the Department of Mathematics, Heriot- Watt University. (2)in this thesis, the word expression is taken to be a mathematical entity which does not contain any comparison operators.