  This article is mostly a writeup of two talks, the first given in the Besse
Seminar at the Ecole Polytechnique in 1998 and the second given at the 2000
International Congress on Differential Geometry in memory of Alfred Gray in
Bilbao, Spain.
  It begins with a discussion of basic geometry of almost complex 6-manifolds.
In particular, I define a 2-parameter family of intrinsic first-order
functionals on almost complex structures on 6-manifolds and compute their
Euler-Lagrange equations.
  It also includes a discussion of a natural generalization of holomorphic
bundles over complex manifolds to the almost complex case. The general almost
complex manifold will not admit any nontrivial bundles of this type, but there
is a large class of nonintegrable almost complex manifolds for which there are
such nontrivial bundles. For example, the standard almost complex structure on
the 6-sphere admits such nontrivial bundles.
  This class of almost complex manifolds in dimension 6 will be referred to as
quasi-integrable. Some of the properties of quasi-integrable structures (both
almost complex and unitary) are developed and some examples are given. However,
it turns out that quasi-integrability is not an involutive condition, so the
full generality of these structures in Cartan's sense is not well-understood.
