We outline the many physics opportunities offered by a multi-purpose fixed-target experiment using the proton and lead-ion beams of the LHC extracted by a bent crystal. In a proton run with the LHC 7 TeV beam, one can analyze pppp, pdpd and pApA collisions at center-of-mass energy View the MathML sourcesNN≃115GeV and even higher using the Fermi motion of the nucleons in a nuclear target. In a lead run with a 2.76 TeV-per-nucleon beam, View the MathML sourcesNN is as high as 72 GeV. Bent crystals can be used to extract about 5×108 protons/s; the integrated luminosity over a year reaches 0.5 fb−1 on a typical 1 cm long target without nuclear species limitation. We emphasize that 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 distributions of the proton and the neutron can be accessed at large xx and even at xx larger than unity in the nuclear case. Single diffractive physics and, for the first time, the large negative-xFxF domain can be accessed. The nuclear target-species versatility provides a unique opportunity to study nuclear matter versus the features of the hot and dense matter formed in heavy-ion collisions, including the formation of the quark-gluon plasma, which can be studied in PbAPbA 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 parton distributions. We also emphasize the potential offered by pApA ultra-peripheral collisions where the nucleus target AA is used as a coherent photon source, mimicking photoproduction processes in epep collisions. Finally, we note that WW and ZZ bosons can be produced and detected in a fixed-target experiment and in their threshold domain for the first time, providing new ways to probe the partonic content of the proton and the nucleus.