We outline the many physics opportunities offered by a multi-purpose fixed-target experiment using the LHC proton and Pb beams extracted by a bent crystal. In a proton run with the LHC 7-TeV beam, one can analyze pp, pd and pA collisions at sqrt(s_NN)~115 GeV and even higher using the Fermi motion in a nuclear target. In a Pb run with a 2.76 TeV-per-nucleon beam, sqrt(s_NN) is as high as 72 GeV. Bent crystals can be used to extract about 5x10^8 protons/s; the integrated luminosity over a year reaches 0.5fb-1 on a typical 1 cm-long target without species limitation. Such an extraction mode does not alter the performance of the collider experiments at the LHC. By instrumenting the target-rapidity region, gluon and heavy-quark proton and neutron PDFs can be accessed at large x and even at x larger than 1 in the nuclear case. Single diffractive physics and, for the first time, the large negative-xF domain can be accessed. The nuclear target-species versatility provides a unique opportunity to study nuclear matter vs. the features of the hot and dense matter formed in heavy-ion collisions, which can be studied in PbA collisions over the full range of target-rapidity domain with a large variety of nuclei. The polarization of hydrogen and nuclear targets allows an ambitious spin program, including measurements of the QCD lensing effects which underlie the Sivers single-spin asymmetry, the study of transversity distributions and possibly of polarized PDFs. We also emphasize the potential offered by pA ultra-peripheral collisions where the nucleus target A is used as a coherent phot Show Partial Abstract