  It is shown that $n$ points and $e$ lines in the complex Euclidean plane
${\mathbb C}^2$ determine $O(n^{2/3}e^{2/3}+n+e)$ point-line incidences. This
bound is the best possible, and it generalizes the celebrated theorem by
Szemer\'edi and Trotter about point-line incidences in the real Euclidean plane
${\mathbb R}^2$.
