The wave-particle dualism becomes very obvious in matter wave interference experiments. Neutron interferometers based on wave front and amplitude division have been developed in the past. Most experiments have been performed with the perfect crystal neutron interferometer, which provides widely separated coherent beams allowing new experiments in the field of fundamental, nuclear, and solid-state physics. A nondispersive sample arrangement and the difference of stochastic and deterministic absorption have been investigated. In case of a deterministic absorption process the attenuation of the (...) interference pattern is proportional to the beam attenuation, whereas in case of stochastic absorption it is proportional to the square root of the attenuation. This permits the formulation of Bell-like inequalities which will be discussed in detail. The verification of the4π symmetry of spinors and of the quantum mechanical spin-superposition experiment on a macroscopic scale are typical examples of interferometry in spin space. These experiments were continued with two resonance coils in the beams, where the results showed that coherence persists, even if an energy exchange between the neutron and the resonator system occurs with certainty. A quantum beat effect was observed when slightly different resonance frequencies were applied to both beams. In this case, the extremely high energy sensitivity of2.7×10 −19 eV was achieved. This effect can be interpreted as a magnetic Josephson-effect analog. Phase echo systems and experiments with pulsed beams show how interference phenomena can be made visible by a proper beam handling inside and behind the interferometer. All the results obtained until now are in agreement with the formalism of quantum mechanics but stimulate the discussion about the interpretation of this basic theory. (shrink)