  Gromov's famous non-squeezing theorem (1985) states that the standard
symplectic ball cannot be symplectically squeezed into any cylinder of smaller
radius. Does there exist an analogue of this result in contact geometry? Our
main finding is that the answer depends on the sizes of the domains in
question: We establish contact non-squeezing on large scales, and show that it
disappears on small scales. The algebraic counterpart of the (non)-squeezing
problem for contact domains is the question of existence of a natural partial
order on the universal cover of the contactomorphisms group of a contact
manifold. In contrast to our earlier beliefs, we show that the answer to this
question is very sensitive to the topology of the manifold. For instance, we
prove that the standard contact sphere is non-orderable while the real
projective space is known to be orderable. Our methods include a new embedding
technique in contact geometry as well as a generalized Floer homology theory
which contains both cylindrical contact homology and Hamiltonian Floer
homology. We discuss links to a number of miscellaneous topics such as topology
of free loops spaces, quantum mechanics and semigroups.
  An erratum is attached whose purpose is to is to correct a number of
inconsistencies in the main paper. These are related to the grading of
generalized Floer homology and do not affect formulations and proofs of the
main results of the paper.
