In this thesis we deal with the problem of state estimation. The difficulty therein lies in the high dimension of the system which makes the simulation itself very costly even for linear models. Considering the linear damped wave equations, we apply model order reduction techniques in order to reduce this computational effort. We develop a-posteriori error bounds for the reduced models, which can be evaluated efficiently. Furthermore, we consider our reduced order models in combination with state estimation methods to reduce the computational effort. We extend our error bounds to account for the whole state estimation error and we compare our approach of using a reduced order model with other ideas from literature. Finally, modeling a gas pipeline network with Euler-type equations, we present two model hierarchies to efficiently estimate the network's state. While the first approach is based on a linearized and dimension-reduced surrogate model, the second one uses nonlinear model order reduction methods. We develop a state estimation process suitable for implicit time integrators to handle the nonlinear estimation problem. The two hierarchies are compared with respect to approximation quality and speedup.