This monograph is a set of course notes written for the 21st Brazilian Mathematics Colloquium held at IMPA in July 1997. It gives an overview of the field of self-validated numerics-computation models in which approximate results are automatically provided with guaranteed error bounds. We focus, in particular, on two such models: interval arithmetic and affine arithmetic.

Interval arithmetic (IA) was developed in the 1960's by Ramon E. Moore. IA is the simplest and most effcient of all validated numerics models, and, not surprisingly, the most widely known and used. After two decades of relative neglect, IA has been enjoying a strong and steady resurgence, driven largely by its successful use in all kinds

of practical applications. We are confident that many readers of this monograph will find IA to be a useful tool in their own work as well.

Affine arithmetic (AA) is a more complex and expensive computation model, designed to give tighter and more informative bounds than IA in certain situations where the latter is known to perform poorly. The AA

model was proposed and developed recently by the authors though a similar model had been developed in 1975

by E. R. Hansen. Apart from its usefulness for certain special applications, AA is being presented here as an example of the many topics for research that are still unexplored in the field of self-validated numerical methods.