  This paper describes the physics case for a new fixed target facility at CERN
SPS. The SHiP (Search for Hidden Particles) experiment is intended to hunt for
new physics in the largely unexplored domain of very weakly interacting
particles with masses below the Fermi scale, inaccessible to the LHC
experiments, and to study tau neutrino physics. The same proton beam setup can
be used later to look for decays of tau-leptons with lepton flavour number
non-conservation, $\tau\to 3\mu$ and to search for weakly-interacting sub-GeV
dark matter candidates. We discuss the evidence for physics beyond the Standard
Model and describe interactions between new particles and four different
portals - scalars, vectors, fermions or axion-like particles. We discuss
motivations for different models, manifesting themselves via these
interactions, and how they can be probed with the SHiP experiment and present
several case studies. The prospects to search for relatively light SUSY and
composite particles at SHiP are also discussed. We demonstrate that the SHiP
experiment has a unique potential to discover new physics and can directly
probe a number of solutions of beyond the Standard Model puzzles, such as
neutrino masses, baryon asymmetry of the Universe, dark matter, and inflation
