  A (flat) affine $3$-manifold is a $3$-manifold with an atlas of charts to an
affine space $\mathbb{R}^3$ with transition maps in the affine transformation
group $\mathrm{Aff}(\mathbb{R}^3)$. We will show that a connected closed affine
$3$-manifold is either an affine Hopf $3$-manifold or decomposes canonically to
concave affine submanifolds with incompressible boundary, toral
$\pi$-submanifolds and $2$-convex affine manifolds, each of which is an
irreducible $3$-manifold. It follows that if there is no toral
$\pi$-submanifold, then $M$ is prime. Finally, we prove that if a closed affine
manifold is covered by a domain in $\mathbb{R}^{n}$, then $M$ is irreducible or
is an affine Hopf manifold.
