  We classify the full set of convex central configurations in the Newtonian
four-body problem. Particular attention is given to configurations possessing
some type of symmetry or defining geometric property. Special cases considered
include kite, trapezoidal, co-circular, equidiagonal, orthodiagonal, and
bisecting-diagonal configurations. Good coordinates for describing the set are
established. We use them to prove that the set of four-body convex central
configurations with positive masses is three-dimensional, a graph over a domain
$D$ that is the union of elementary regions in $\mathbb{R}^{+^3}$.
