  In this paper, we study the geometric aspects of ball packings on
$(M,\mathcal{T})$, where $\mathcal{T}$ is a triangulation on a 3-manifold $M$.
We introduce a combinatorial Yamabe invariant $Y_{\mathcal{T}}$, depending on
the topology of $M$ and the combinatoric of $\mathcal{T}$. We prove that
$Y_{\mathcal{T}}$ is attainable if and only if there is a constant curvature
packing, and the combinatorial Yamabe problem can be solved by minimizing
Cooper-Rivin-Glickenstein functional. We then study the combinatorial Yamabe
flow introduced by Glickenstein \cite{G0}-\cite{G2}. We first prove a small
energy convergence theorem which says that the flow would converge to a
constant curvature metric if the initial energy is close in a quantitative way
to the energy of a constant curvature metric. We shall also prove: although the
flow may develop singularities in finite time, there is a natural way to extend
the solution of the flow so as it exists for all time. Moreover, if the
triangulation $\mathcal{T}$ is regular (that is, the number of tetrahedrons
surrounding each vertex are all equal), then the combinatorial Yamabe flow
converges exponentially fast to a constant curvature packing.
