We consider Hilbert transforms along curves
$\Gamma$ on the Heisenberg group. In particular, necessary and
sufficient conditions for the $L^2$-boundedness are given for
$\Gamma(t)=(t,\gamma(t),0)$ when $\gamma$ is even or odd and
convex on ${\Bbb R}^{+}$. 1991 Mathematics Subject Classification: 42B20,42B25.