In this paper, we aim at presenting the most recent results achieved in
the theory of pseudoprime numbers. First of all, we make a list of
all pseudoprime varieties existing so far: This includes
Lucas-pseudoprimes and the generalization to sequences generated by
integer polynomials modulo <I>N</I>, elliptic pseudoprimes.
We discuss the making of
tables and the consequences on the design of very fast primality
algorithms for small numbers. Then, we describe the
recent work of Alford, Granville and Pomerance, in which they prove
that there exists an infinite number of Carmichael numbers. We discuss
also the potential applications of their work to other classes of numbers.
Then, we describe Arnault's work and the counterattack of Davenport:
we explain how to generate pseudoprimes that defeat the primality
routines of some computer algebra system such as {\sc Axiom} and {\sc
Maple}. In another section, we present some recent work of Atkin on
generalizing Miller-Rabin's algorithm.