Physics reports 808, 1 - 74 (2019). doi:10.1016/j.physrep.2019.02.002, Free-electron lasers produce extremely brief, coherent, and bright laser-like photon pulses that allow to image matter at atomic resolution and at timescales faster than the characteristic atomic motions. In pulses of about 50 femtoseconds duration they provide as many photons as one gets in 1 s from modern storage ring synchrotron radiation facilities. FLASH, the Free-Electron Laser at DESY in Hamburg was the first FEL in the XUV/soft X-ray spectral range, started operation as a user facility in summer 2005, and was for almost 5 years the only short wavelength FEL facility worldwide. Hence, most of the technological developments as well as the scientific experiments performed by the user community were new and unique as outlined below. FLASH was driving FEL science and technology and paved the way for many new ideas. Because of using a linear accelerator in superconducting RF technology FLASH combines the extreme peak brightness characteristic for FELs with very high average brightness. It also was the prototype for the European XFEL located in the Hamburg metropolitan area, which started user operation in summer 2017.The present review provides an overview of the progress made with accelerator science and technology at FLASH for the production of stable beams of well characterized electron pulses, reduction of the pulse jitter to the femtosecond level, generation of ultra-short photon pulses, adequate synchronization of the machine parameters with the experiment, and demonstrating advanced FEL schemes using variable gap undulators. Much of this was done in the very exciting early days of FEL science when it was even not clear if the FEL concept could be realized for X-rays. The development and the operation of the FLASH user facility is described, as well as the techniques developed to make use of the new type of X-ray beams including photon beam diagnostics and damage studies of the optical elements. The review emphasizes breakthrough experiments which demonstrated that many of the ideas collected in the world-wide discussion of the scientific case of free-electron lasers could indeed be realized and they often produced unexpected results. The first experiment on Coulomb explosion of Xe clusters performed in 2002 was a clear demonstration of the feasibility of experiments with free-electron laser beams and opened a lively discussion in the atom, molecular and optical physics community (AMO).Time resolved single-shot single-particle imaging, summarized in the slogan “Take movies instead of pictures”, was one of the most popular science drivers for the construction of free-electron X-ray lasers. As a first step in this direction experiments using a highly focused beam of FLASH demonstrated that pictures of 2 dimensional objects could be reconstructed from single-shot single-particle diffraction patterns. Explosion dynamics of nano-size particles hit by an intense FEL pulse were studied. This method, called “diffraction before destruction”, is now very successfully applied with hard X-rays and, to a large extent, solves the radiation damage problem in structural biology. A long term goal is to determine the 3 dimensional structure of a large molecule from a single-shot diffraction pattern. Along these lines the 3D architecture of free Ag nanoparticles could be determined from one diffraction pattern only using soft X-rays from FLASH.To understand light–matter interactions in this new parameter space a number of pioneering AMO experiments have been performed including non-linear interactions in atoms, molecules and clusters. Multiphoton photoionization processes in the presence of intense optical fields have been studied, as well as photo-absorption of XUV photon energies on molecular ions important for astrophysics. The nature of formation and breaking of molecular bonds was investigated in VUV pump–VUV probe experiments using a reaction microscope and a specific delay line. As an example the process of ultrafast isomerization of acetylene molecules C$_2$H$_2$ triggered by single photon excitation has been studied. The structural changes during the isomerization process were visualized and an isomerization time of 52 +/- 15 fs was found.Clusters of variable size, which can be produced routinely, allow distinguish between inter- and intra-atomic effects and are considered model systems for the investigation of light–matter interactions in multi-atom objects. As an example such experimental studies provided instructive data for benchmarking theoretical models describing cluster ionization in intense short-wavelength laser pulses. The combination of single-shot single-particle imaging for determination of the cluster size with spectroscopy was crucial for success of these experiments. The investigations could later be extended to very large Xe clusters providing new insights into the nanoplasma formation and explosion dynamics of such large systemsFrom early on, studies of high energy density plasmas and warm dense matter have been one of the most prominent research fields in building the scientific case for X-ray free-electron lasers. A good understanding of this complex regime between cold solids and hot dilute plasmas is important for high pressure studies, applied materials studies, inertial fusion, and planetary interiors. With the first observation of saturable absorption of an L-shell transition in Aluminum and pioneering studies of warm dense hydrogen FLASH kicked off research of matter in extreme conditions with free-electron lasers.In condensed matter experiments the emphasis is not so much on the peak power of the FEL beam and extreme focusing, but on beam properties like polarization and pulse duration. The sample has to stay intact in the beam over hours and the number of photons per pulse impinging on the sample has to be limited to avoid space charge effects. After demonstrating the possibility to record single-shot resonant magnetic scattering images with FELs the first time-resolved demagnetization study using a pump–probe approach with an IR-pump pulse and an XUV probe pulse to record a resonant magnetic scattering pattern as a function of pump–probe delay was also performed at FLASH.Free-electron lasers offer the possibility to extend the well-established X-ray spectroscopic techniques for the investigation of the static electronic structure of matter to probing the evolution of the electronic structure in the time domain after controlled excitation. At FLASH first time resolved core level photoemission (TR-XPS) experiments have been performed which are element specific and provide information on the dynamics of the local charge state around a specific center. Using 198 eV photons in a surface study at Ir single crystals it was possible to separate surface and bulk contributions in the Ir 4f levels with sufficient instrumental resolution. Time and angular resolved photoelectron spectroscopy (TR-ARPES) is a very powerful tool to study non-equilibrium electron dynamics of condensed matter systems, since it offers the possibility to follow the dynamics of the full band structure of a material. In another pioneering experiment the photo-induced dynamics of the Mott insulator 1T-TaS$_2$ was studied at FLASH by investigating the dynamics of the Ta 4f photoemission. The formation of a commensurate charge density wave (CCDW) leads to a splitting of the Ta 4f level which decreases first on a sub-picosecond time scale due to electronic melting of the CCDW and afterwards on a picosecond lifetime due to electron–phonon coupling. This leads to transfer of energy from the electronic system to the lattice and a partial melting of the periodic lattice distortions accompanying the periodic charge arrangement in the CCDW phase.In materials science X-ray absorption and emission spectroscopy are among the most powerful spectroscopies to study the electronic structure of matter. The wavelength of the radiation is scanned over certain element specific resonances which at FLASH 1 can only be done by scanning the electron energy. This is time consuming and makes the experiments difficult. Nevertheless, the first time-resolved X-ray emission spectroscopy (XES) experiment was done at FLASH 1 in order to study non-thermal melting of a silicon sample. From a comparison of the observed valence electronic structure at different times after the photoexcitation it became clear that in the melting process in the first few ps a non-equilibrium low density liquid state is reached. The existence of such a metastable low density liquid state had been postulated for many systems that show tetragonal bonding in the crystalline phase like water for example, but spectroscopically the time-resolved silicon XES data taken at FLASH verified its existence for the first time. FLASH 2 has tunable undulators and it was demonstrated that scanning of the wavelength is very easy there., Published by Elsevier Science, North-Holland, Amsterdam [u.a.]