  The Alexander-Hirschowitz theorem says that a general collection of $k$
double points in ${\bf P}^n$ imposes independent conditions on homogeneous
polynomials of degree $d$ with a well known list of exceptions. Alexander and
Hirschowitz completed its proof in 1995, solving a long standing classical
problem, connected with the Waring problem for polynomials. We expose a
self-contained proof based mainly on previous works by Terracini, Hirschowitz,
Alexander and Chandler, with a few simplifications. We claim originality only
in the case $d=3$, where our proof is shorter. We end with an account of the
history of the work on this problem.
