  The asymptotic behaviour of vacuum Bianchi models of class A near the initial
singularity is studied, in an effort to confirm the standard picture arising
from heuristic and numerical approaches by mathematical proofs. It is shown
that for solutions of types other than VIII and IX the singularity is velocity
dominated and that the Kretschmann scalar is unbounded there, except in the
explicitly known cases where the spacetime can be smoothly extended through a
Cauchy horizon. For types VIII and IX it is shown that there are at most two
possibilities for the evolution. When the first possibility is realized, and if
the spacetime is not one of the explicitly known solutions which can be
smoothly extended through a Cauchy horizon, then there are infinitely many
oscillations near the singularity and the Kretschmann scalar is unbounded
there. The second possibility remains mysterious and it is left open whether it
ever occurs. It is also shown that any finite sequence of distinct points
generated by iterating the Belinskii-Khalatnikov-Lifschitz mapping can be
realized approximately by a solution of the vacuum Einstein equations of
Bianchi type IX.
