  Algebra bundles, in the strict sense, appear in many areas of geometry and
physics. However, the structure of an algebra is flexible enough to vary
non-trivially over a connected base, giving rise to a structure of a weak
algebra bundle. We will show that the notion of a weak algebra bundle is more
natural than that of a strict algebra bundle. We will give necessary and
sufficient conditions for weak algebra bundles to be locally trivial. The
collection of non-trivial associative algebras of a fixed dimension forms a
projective variety, called associator varieties. We will show that these
varieties play the role the Grassmannians play for principal $O(n)$-bundles.
