Neutron experiments provide a unique opportunity in some cases to probe extra short-range interactions. We will consider the most sensitive experiments. Spin-dependent (axion-like) interactions in the distance range of 10^(-6)-10^(-3) m are best constrained from the observation of the gravitationally-bound quantum states of neutrons. Further significant progress is expected in the new-generation experiment GRANIT on resonance transitions between the gravitationally bound quantum states of neutrons. Spin-independent short range forces in the distance range of 10^(-12)-10^(-8) m are best constrained from the neutron scattering experiments. One could probably extend the distance range for such forces to sub-microns using centrifugal quantum states of neutrons, and/or to microns using GRANIT.