The nonlinear damped wave equation \[ u_{tt}+\beta u_t-\Delta u+f(u)=0 \] is considered on a bounded domain \(\Omega\subset {\mathbb R}^n\) imposing Dirichlet boundary conditions. For the nonlinearity it is assumed that \(\liminf_{| u| \to\infty}f(u)/u>-\lambda_1\), with \(\lambda_1\) the first eigenvalue of \(-\Delta\). In addition, the growth condition \(| f(u)| \leq C(1+| u| ^{n/(n-2)})\) is supposed if \(n\geq 3\), whereas \(f\) may grow exponentially for \(n=2\). The main result of the paper asserts that the equation has a connected global attractor in \(H_0^1(\Omega)\times L^2(\Omega)\), identifying \(u\) with \((u, u_t)\). It is further shown that for each global orbit in the attractor the \(\alpha\)- resp.~\(\omega\)-limit set is a connected subset of the critical points of the Lyapunov functional \(V(u, u_t)=\int_\Omega\{(1/2)u_t^2+(1/2)| \nabla u| ^2+F(u)\}\,dx\), where \(F'=f\). If the set of critical points is totally disconnected, then the solutions do not only approach the attractor as a set, but they converge to an individual critical point as \(t\to\pm\infty\). The proofs rely on the application of suitable abstract results concerning the existence of attractors.