  We construct three kinds of complete embedded minimal surfaces in $\Bbb
H^2\times \Bbb R$. The first is a simply connected, singly periodic, infinite
total curvature surface. The second is an annular finite total curvature
surface. These two are conjugate surfaces just as the helicoid and the catenoid
are in $\mathbb R^3$. The third one is a finite total curvature surface which
is conformal to $\mathbb S^2\setminus\{p_1,...,p_k\}, k\geq3.$
