This chapter is devoted to the arithmetic Operations, essentially addition, multiplication, exponentiation, division, gcd calculation and evaluation, on the basic algebraic domains. The algorithms for these basic domains are those most frequently used in any Computer algebra system. Therefore the best known algorithms, from a computational point of view, are presented. The basic domains considered here are the rational integers, the rational numbers, integers modulo *m*, Gaussian integers, polynomials, rational functions, power series, finite fields and *P-*adic numbers. Bounds on the maximum, minimum and average Computing time (*t*^{+}, *t*^{-}, *t**) for the various algorithms are given.