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