  We consider the dynamics towards the initial singularity of Bianchi type IX
vacuum and orthogonal perfect fluid models with a linear equation of state.
Surprisingly few facts are known about the `Mixmaster' dynamics of these
models, while at the same time most of the commonly held beliefs are rather
vague. In this paper, we use Mixmaster facts as a base to build an
infrastructure that makes it possible to sharpen the main Mixmaster beliefs. We
formulate explicit conjectures concerning (i) the past asymptotic states of
type IX solutions and (ii) the relevance of the Mixmaster/Kasner map for
generic past asymptotic dynamics. The evidence for the conjectures is based on
a study of the stochastic properties of this map in conjunction with dynamical
systems techniques. We use a dynamical systems formulation, since this approach
has so far been the only successful path to obtain theorems, but we also make
comparisons with the `metric' and Hamiltonian `billiard' approaches.
