The papers in this special section examine the concept of computer arithmetic. Many services offered in the palm of our hand by today’s computing devices were undreamt of twenty years ago, and we probably don’t envision what services will be enabled twenty years from now. We even can’t be sure of the technology they will use, if good old silicon integration is no longer able to sustain Moore’s law. However, one can be confident that there will be computers, and that these computers will compute, and that at the core of these computations there will be adders, multipliers, elementary functions and other core arithmetic primitives. Computer arithmetic is the art of designing and using these core arithmetic primitives. It studies the representation of numbers in computers, and the transformation of these “machine numbers”. With the abacus and early mechanical calculators, it actually predates the computing era. Computer arithmetic has accompanied the evolutions of technology (from relays to vacuum tube to transistors and integrated circuits). It has also adapted to the evolution of applications: scientific computing, digital signal processing, cryptography, or machine learning use different kinds of numbers and operations. Formal proofs involving computer arithmetic components have become a major concern of many other applications.