Charge exchange between ions and neutral atoms or molecules has been accepted at an important soft producing process in our solar system. By extension, charge exchange may contribute to the X-ray emission of circumstellar material. It may also produce X-ray emission at the boundaries of supernova ejecta and star burst galaxies, or whenever hot plasma collides with neutral matter. X-ray spectra of K-shell and L-shell ions formed by charge exchange have now been studied in a variety of laboratory settings. These experiments have shown several characteristic features of line formation by charge exchange when compared to the X-ray emission produced by electron-impact excitation, e.g., enhancement emission of forbidden lines and of lines from levels with high principal quantum number. They have also shown a dependence on the interaction gas and on the energy of the ion-neutral collision. Moreover, the transfer of multiple electrons is typically preferred, provided the donor molecules or atoms have multiple valence-shell electrons. The laboratory measurements are in qualitative agreement with theory. However, the details of the observed X-ray spectra, especially those recorded with high spectral resolution, can differ substantially from predictions, especially for spectra produced at collision velocities equal to or lower than those found in thermal plasmas or produced with neutral gases other than atomic hydrogen. Puzzling discrepancies can be noted, such as enhanced emission from an upper level with the 'wrong' principal quantum number. Even more puzzling is a recent experiment in which two, co-mixed bare ion species of similar atomic number produce very different Lyman series emission upon charge exchange with a given neutral gas, defying both theoretical predictions and empirical scaling. Laboratory measurements have also shown that some of the characteristic features of charge exchange can be reproduced by radiative electron capture, i.e., by capture of a continuum electron. This process dominates in cold, photo-ionized plasmas, or when hot and cold plasmas mix. Work was performed under the auspices of the DOE by LLNL under contract DE-AC52-07NA27344 and supported in part by an award from NASA's APRA program.