We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincar�� metrics (i.e., complete metrics of constant negative curvature) by solving the equation $��u - e^{2u} = K_0(z)$ on general open surfaces. A few other topics are discussed, including boundary behavior of the conformal factor $e^{2u}$ giving the Poincar�� metric when the Riemann surface has smoothly bounded compact closure, and also a curvature equation proof of Koebe's disk theorem.

26 pages