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.
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