Since 1976 the raise of public key cryptosystems (such as RSA) has led to the development of Computational Number Theory. Its aim is to solve simple problems such as factoring numbers into primes, as quickly as possible. In this article, we survey the algorithmic solutions given to three problems: how to prove a number prime; how to factor a number; how to compute discrete logarithms in finite fields. We also stress the relationships between the high level of mathematics involved (elliptic curves, number fields, etc.) and the use of new computational means (parallel machines, networks of workstations).