When Rudolf Mößbauer received his Nobel Prize in Physics 1961, he was only 31 years old. Since the Lindau meetings had already started some ten years earlier, he could look forward to many invitations, not only to the meetings in physics, but to all meetings. In the beginning he chose the physics ones and mostly lectured on what is named after him, the Mößbauer effect, and its applications. The present lecture is his second at Lindau and mainly concerns the applications of this effect in the general area of solid-state physics. Most atomic nuclei can absorb and emit high-energy electromagnetic radiation of extremely well defined frequency, so called characteristic gamma rays. But the frequencies are generally shifted due to the Doppler effect, which becomes active because of the nuclear recoil, and so the frequencies become smeared. What Mößbauer found was that if the atoms are bound in crystal lattices, the nuclear recoil energy can be shared among all the atoms of the crystal. This means that the Doppler effect becomes negligible and the frequencies well defined. Using one crystal for emission and another one for absorption, it becomes possible to measure extremely small frequency shifts. These could derive from moving the whole crystal with velocities down to millimetres per hour or even from changes in the gravitational potential between emitter and absorber (according to the general theory of relativity). The main use of the method, as described by Mößbauer, is to measure the so-called hyperfine effects. These are small shifts of the gamma frequencies that depend on the local fields at the atomic nuclei. These fields can derive from the electronic and magnetic properties of the electron clouds and the shifts also depend on the shape of the atomic nuclei. So measuring these effects can give important information both on the electronic structure of the atoms in the crystals and on their nuclei.