  Let $V$ be a finite dimensional complex vector space and $W\subset \GL(V)$ be
a finite complex reflection group. Let $V^{\reg}$ be the complement in $V$ of
the reflecting hyperplanes. A classical conjecture predicts that $V^{\reg}$ is
a $K(pi,1)$ space. When $W$ is a complexified real reflection group, the
conjecture follows from a theorem of Deligne. Our main result validates the
conjecture for duality (or, equivalently, well-generated) complex reflection
groups. This includes the complexified real case (but our proof is new) and new
cases not previously known. We also address a number of questions about
$\pi_1(W\cq V^{\reg})$, the braid group of $W$.
