  Let $V$ be a finite dimensional complex vector space and $W\subseteq \GL(V)$
be a finite complex reflection group. Let $V^{\reg}$ be the complement in $V$
of the reflecting hyperplanes. We prove that $V^{\reg}$ is a $K(\pi,1)$ space.
This was predicted by a classical conjecture, originally stated by Brieskorn
for complexified real reflection groups. The complexified real case follows
from a theorem of Deligne and, after contributions by Nakamura and
Orlik-Solomon, only six exceptional cases remained open. In addition to solving
this six cases, our approach is applicable to most previously known cases,
including complexified real groups for which we obtain a new proof, based on
new geometric objects. We also address a number of questions about $\pi_1(W\cq
V^{\reg})$, the braid group of $W$. This includes a description of periodic
elements in terms of a braid analog of Springer's theory of regular elements.
