Liverpool John Moores University and FRAME recently conducted a research project sponsored by Defra, on the status of alternatives to animal testing with regard to the European Union REACH (Registration, Evaluation and Authorisation of Chemicals) system for the safety testing and risk assessment of chemicals. The project covered all the main toxicity endpoints associated with the REACH system. This paper focuses on the prospects for using alternative methods (both in vitro and in silico) for mutagenicity (genotoxicity) and carcinogenicity testing - two toxicity endpoints, which, together with reproductive toxicity, are of pivotal importance for the REACH system. The manuscript critically discusses well-established testing approaches, and in particular, the requirement for short-term in vivo tests for confirming positive mutagenicity, and the need for the rodent bioassay for detecting non-genotoxic carcinogens. Recently-proposed testing strategies focusing on non-animal approaches are also considered, and our own testing scheme is presented and supported with background information. This scheme makes maximum use of pre-existing data, computer (in silico) and in vitro methods, with weight-of-evidence assessments at each major stage. The need for the improvement of in vitro methods, to reduce the generation of false-positive results, is also discussed. Lastly, ways in which reduction and refinement measures can be used are also considered, and some recommendations are made for future research to facilitate the implementation of the proposed testing scheme.