  In this article, we study the topology and bifurcations of the moduli space
$\mathcal{M}_3$ of cubic Newton maps. It's a subspace of the moduli space of
cubic rational maps, carrying the Riemann orbifold structure
$(\mathbb{\widehat{C}}, (2,3,\infty))$. We prove two results: (1). The boundary
of the unique unbounded hyperbolic component is a Jordan arc and the boundaries
of all other hyperbolic components are Jordan curves. (2).The Head's angle map
is surjective and monotone. The fibers of this map are characterized
completely.
  The first result is a moduli space analogue of the first author's dynamical
regularity theorem \cite{Ro08}. The second result confirms a conjecture of Tan
Lei.
