In thermoelastic systems, the dissipation through heat conduction plays a prominent role for the question of possible exponential stability of the system. Replacing the constitutive law of Fourier by that of Cattaneo(-Maxwell), which is often done to get a hyperbolic system with finite propagation speed of signals, often yields qualitatively similar
results. We report on situations, where qualitative differences appear. Moreover, we discuss delay equations and mention hyperbolic Navier-Stokes equations.