Many research directions and applications share the common objective of computing an estimate for the unknown state of a dynamic system. The Kalman filter has evolved to a well-established concept for estimating the system's state on the basis of noisy measurements. However, a major limitation in applying the Kalman filter is less the assumption of Gaussian distributions that characterize the involved uncertainties but more generally the use of a purely probabilistic model. This model can be too specific and too restrictive when different types of uncertainties have to be dealt with. Important examples include uncertainties originating from unknown systematic errors, quantization and linearization errors, or disturbances related to an unreliable communication network. The Kalman filter and the majority of well-known estimation techniques require a purely probabilistic modeling of uncertainties and thus can render the solving of many estimation problems more difficult. In the course of preliminary work, the Kalman filter algorithm has been extended in order to complement the common stochastic uncertainty characterization by a non-stochastic model that relies on set-membership uncertainty descriptions. This uncertainty model is particularly tailored to unknown but bounded disturbances. Unlike random errors, which are characterized by probability distributions, a set-membership model does not require any specific error behavior to be pinpointed - except for boundedness. This combined approach, which is referred to as LM²MSE estimator (LM²MSE: Linear Min-Max Mean Squared Error) in this proposal, enables a simultaneous treatment of stochastic as well as unknown but bounded uncertainties and remains simple to implement.This research project primarily aims at extending the common purely stochastic approach to state estimation by a set-membership uncertainty model. The further analysis and development of the LM²MSE concept is subdivided into three aspects to be addressed. In line with the first aspect, important properties of the LM²MSE filter are to be pointed out. These include, in particular, convergence properties and an assessment of its estimation quality. This aspect also covers important questions concerning the filter's application to control problems. A comparison with other approaches that allow for a simultaneous treatment of different types of uncertainties lies in the focus of the second aspect. The third aspect is dedicated to important further developments in order to tackle continuous-time as well as nonlinear state estimation problems and to allow for distributed computations of state estimates. The results of this research project are expected to spur a new perspective on the treatment of measurement uncertainties and to provide simpler solutions to many estimation and control problems. Accompanying application and case studies highlight the benefits of this new concept.