This text contains 22 lectures presented at the final conference of the German research programme "Algorithmic Number Theory and Algebra 1991-1997", sponsored by the Deutsche Forschungsgemeinschaft. The purpose of this research programme and the meeting was to bring together developers of computer algebra software and researchers using computational methods to gain insight into experimental problems and theoretical questions in algebra and number theory. The book gives an overview on algorithmic methods and results obtained during this period mainly in algebraic geometry, and group and representation theory. Some of the articles illustrate the current state of the computer algebra systems developed with support from the research programme, for example KANT and LIDiA for algebraic number theory, SINGULAR, REDLONG and INVAR for commutative algebra and invariant theory respectively, and GAP, SYSYPHOS and CHEVIE for group and representation theory.