Crystals are the elementary constituents of Laue lenses, an emerging technology which could allow the realization of a space-borne telescope 10-100 times more sensitive than existing ones, in the 100 keV-1.5 MeV energy range. This paper addresses the development of efficient crystals for the realization of a Laue lens. In the theoretical part, 35 candidate crystals, both pure and two-component crystals, are considered. Their peak reflectivity at 100, 500 keV and 1 MeV is calculated assuming they are mosaic crystals. It is found that, by careful selection of crystals, it is possible to achieve a reflectivity above 30% over the whole energy range, and even up to 40% in the lower part of the energy range. In the experimental part, three different materials (Si1-xGex with a gradient of composition, mosaic Cu and Au) have been measured at both ESRF and ILL using highly monochromatic beams ranging from 300 to 816 keV. The aim was to check their homogeneity, quality and angular spread (mosaicity). These crystals have shown outstanding performance, such as reflectivity up to 31% at similar to 600 keV (Au) or 60% at 300 keV (SiGe) and angular spread as low as 15 arcsec for Cu, fulfilling very well the requirements for a Laue lens application. An unexpected finding is that there are important discrepancies with Darwin's model when a crystal is measured using various energies.