The book deals with computer arithmetic in a more general sense than usual. Advanced computer arithmetic requires that all computer approximations of arithmetic operations – in particular those in the usual vector and matrix spaces – differ from the correct result by at most one rounding. The implementation of advanced computer arithmetic by fast hardware is examined in the book. The new expanded computational capability is gained at modest cost. It increases both the speed of a computation and the accuracy of the computed result. With it fast multiple precision arithmetic can be easily provided. All this strongly supports the case for implementing advanced computer arithmetic on every CPU. The book also shows that on superscalar processors interval operations can be made as fast as simple floating-point operations with only very modest additional hardware costs.