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Neutron scattering is a well-established tool for the investigation of the static and dynamic properties of condensed matter systems over a wide range of spatial and temporal scales. Many studies of high interest, however, can only be performed on small samples and typically require elaborate environments for variation of parameters such as temperature, magnetic field and pressure. To improve the achievable signal-to-background ratio, focusing devices based on elliptic or parabolic neutron guides or Montel mirrors have been implemented. Here we report an experimental demonstration of a nested mirror optics (NMO), which overcomes some of the disadvantages of such devices. While even simpler than the original Wolter design, our compact assembly of elliptic mirrors images neutrons from a source to a target, minimizing geometric aberrations, gravitational effects and waviness-induced blurring. Experiments performed at MIRA at FRM-II demonstrate the expected focusing properties and a beam transport efficiency of 72 % for our first prototype. NMO seem particularly well-suited to i) extraction of neutrons from compact high-brilliance neutron moderators, ii) general neutron transport, and iii) focusing and polarizing neutrons. The phase space of the neutrons hitting a sample can be tailored on-line to the needed experimental resolution, resulting in small scattering backgrounds. As additional benefits, NMO situated far away from both the moderator and the sample are less susceptible to radiation damage and can easily be replaced. NMO enable a modular and physically transparent realization of beam lines for neutron physics similar to setups used in visible light optics.