In this paper the authors present a nice contribution to the theory of the quasihyperbolic metric for domains in \({\mathbb{R}}^ n\). For a domain D in \({\mathbb{R}}^ n\) the quasihyperbolic distance \(k_ D(x,y)\) is obtained from the generalized Riemannian metric \(| dx| /d(x,\partial D)\), where d(x,\(\partial D)\) is the Euclidean distance from x to the boundary of D. This metric and distance have numerous applications in geometric function theory. The authors begin by discussing general results in \({\mathbb{R}}^ n\) and contrasting them with the special case \(n=2\). For example, the sectional curvature can be positive, negative or zero if \(n\geq 3\), while the generalized Gaussian curvature is always nonpositive when \(n=2\). They establish the surprising result that an isometry \(f:(D_ 1,k_ 1)\to (D_ 2,k_ 2)\) of two domains in \({\mathbb{R}}^ n\) (n\(\geq 2)\) relative to the quasihyperbolic distance must be a conformal mapping and so a Möbius transformation when \(n\geq 3\). Also, they determine the quasihyperbolic geodesics of the unit ball and note they them seem to have little geometric significance in contrast to the hyperbolic geodesics. The remainder of the paper is devoted to domains in \({\mathbb{R}}^ 2\). In this context the authors show that two elementary comparison principles together with the explicit computation of the curvature of the quasihyperbolic metric in a few special cases suffice to estimate the curvature in a number of general situations. In particular, they show that the quasihyperbolic metric on a region D is an SK-metric (in the sense of M. Heins) if and only if D is convex.