This paper summarizes the development of a successful project, LAUE, supported by the Italian Space Agency (ASI) and devoted to the development of long focal length (up to 100 m) Laue lenses for hard X-/soft gamma-ray astronomy (80-600 keV). The apparatus is ready and the assembling of a prototype lens petal is ongoing. The great achievement of this project is the use of bent crystals. From measurements obtained on single crystals and from simulations, we have estimated the expected Point Spread Function and thus the sensitivity of a lens made of petals. The expected sensitivity is a few ×10 −8 photons cm −2 s −1 keV −1 . We discuss a number of open astrophysical questions that can settled with such an instrument aboard a free-flying satellite.A significant part of the NuSTAR core program time will be devoted to sensitive studies of extragalactic and Galactic surveys, which will supplement the extended surveys beyond 20 keV already performed 7, 8 or planned to be continued with the ESA INTEGRAL observatory, 9 and with the NASA Swift satellite. 10Evidence of extended matter-antimatter annihilation emission (at 511 keV) from the Galactic Center 11 and of Galactic nucleosynthesis processes 11, 12 has been found. Furthermore, polarization of high energy photons (> 400 keV) emitted from a strong source like the Cygnus X-1 has been clearly measured. 13However, in order to take full advantage of these results, a new generation of focusing telescopes which extend the energy band up to several hundreds of keV is needed. In spite of its high sensitivity, NuSTAR optics are based on multilayer reflectivity and therefore are limited in the energy passband. A two-order of magnitude