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In the present paper we review the main results of a series of recent paperson the non-Abelian Chern-Simons path integral on $M=\Sigma \times S^1$ in theso-called "torus gauge". More precisely, we study the torus gauge fixed versionof the Chern-Simons path integral expressions $Z(\Sigma \times S^1,L)$associated to $G$ and $k \in N$ where $\Sigma$ is a compact, connected,oriented surface, $L$ is a framed, colored link in $\Sigma \times S^1$, and $G$is a simple, simply-connected, compact Lie group. We demonstrate that the torusgauge approach allows a rather quick explicit evaluation of $Z(\Sigma \timesS^1,L)$. Moreover, we verify in several special cases that the explicit valuesobtained for $Z(\Sigma \times S^1,L)$ agree with the values of thecorresponding Reshetikhin-Turaev invariant. Finally, we sketch three differentapproaches for obtaining a rigorous realization of the torus gauge fixed CSpath integral. It remains to be seen whether also for general $L$ the explicitvalues obtained for $Z(\Sigma \times S^1,L)$ agree with those of thecorresponding Reshetikhin-Turaev invariant. If this is indeed the case thenthis could lead to progress towards the solution of several open questions inQuantum Topology.

The non-Abelian Chern-Simons path integral on $M=\Sigma \times S^1$ in the torus gauge: a review