The quasihyperbolic metric \(k_D(x,y)\) is a counterpart of the standard hyperbolic metric on a plane domain on an arbitrary proper subdomain \(D\) of \(\mathbb{R}^n\) [{F. W. Gehring}, \textit{B. P. Palka}, J. Anal. Math. 30, 172-199 (1976; Zbl 0349.30019)]. For \(x\in D\) let \(\delta_D(x)\) be the distance from \(x\) to \(\partial D\). Then \(k_D(x,y)= \inf\int_\gamma \partial_D (x)^{-1}ds\) where the infimum is taken over all curves \(\gamma\) joining \(x\) to \(y\) in \(D\). A domain \(D\) is an \(H\)-domain with a base point \(x_0\) if \(\delta_D (x)\leq\delta_D (x_0)H (k_D(x,x_0))\) where \(H:[0,\infty) \to(0,\infty)\) is a continuous function satisfying \(H(t)\geq e^{-t}\), \(t\geq 0\). The condition makes it simple to estimate the Euclidean arc length of quasihyperbolic geodesics and the basic idea in the paper is to use quasihyperbolic length of the geodesic instead of the Euclidean length. John domains, bounded uniform domains and Hölder domains are examples of \(H\)-domains. The author gives necessary and sufficient conditions in order that an \(H\)-domain is bounded. The plumpness of \(H\)-domains is studied. The author also shows that, under some natural conditions on \(H\), there exists an \(H\)-domain \(D\) with \(\delta_D(x)=H(k_D(x,x_0))\).