A connected Riemannian manifold $M$ has constant vector curvature $\epsilon$, denoted by cvc$(\epsilon)$, if every tangent vector $v \in TM$ lies in a 2-plane with sectional curvature $\epsilon$. By scaling the metric on $M$, we can always assume that $\epsilon = -1, 0$, or $1$. When the sectional curvatures satisfy an additional bound sectional curvature $\leq \epsilon$ or sectional curvature $\geq \epsilon$, we say that $\epsilon$ is an {\it extremal} curvature. In this talk we first motivate the definition and then describe the moduli spaces of cvc$(\epsilon)$ metrics on three manifolds for each case, $\epsilon = -1, 0$, or $1$, under global conditions on $M$. For example, in the case $\epsilon = -1$ is extremal, we show, under the assumption that $M$ has finite volume, that $M$ is isometric to a locally homogeneous manifold. In the case that $M$ is compact, $\epsilon = 1$ is extremal and there are no points in $M$ with all sectional curvatures identically one, we describe the moduli space of cvc$(1)$ metrics in terms of locally homogeneous metrics and the solutions of linear elliptic partial differential equations. Solutions of some nonlinear elliptic equations arise in the proof.