We describe a primality testing algorithm, due essentially
to Atkin, that uses elliptic curves over finite fields and the theory of
complex multiplication. In particular, we explain how the use of class 
fields and genus fields can speed up certain phases of the algorithm. 
We sketch the actual implementation of this test and its use on testing
large primes, the records being two numbers of more than 550 decimal
digits. Finally, we give a precise answer to the question of the reliability
of our computations, providing a <I>certificate of primality</I> for a
prime number.