A Riemann surface $X$ is a connected complex manifold of complex dimension one. A homogeneous space is a manifold with a transitive smooth action of a Lie group. I guess there must be a classification of those Riemann surfaces which are also homogeneous spaces . . . however, I can't seem to find it.
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5$\begingroup$ So you don't require the action to preserve the holomorphic structure? Eventually I think it doesn't matter at least in the closed case (where homogeneous $\Leftrightarrow$ genus $\le 1$). In the nonclosed case, it matters: the quotient of the hyperbolic plane by a loxodromic isometry is homogeneous as smooth manifold, but not as Riemann surface. $\endgroup$– YCorFeb 5, 2019 at 17:39

$\begingroup$ What if I include compact? This is what I am really interested in. $\endgroup$– Pierre DuboisFeb 5, 2019 at 18:07

$\begingroup$ Ok, so I guess yes, I would like that the action preserves the holomorphic structure $\endgroup$– Pierre DuboisFeb 5, 2019 at 19:46
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If you only care about compact (oriented) surfaces, then it is easy to see that the only homogeneous examples are the sphere $S^2=\mathbb C P^1$ and the torus $T^2=S^1\times S^1$. Indeed, if $M$ is a compact homogeneous space, then its Euler characteristic is $\chi(M)\geq0$; so for an oriented surface this implies genus $\leq1$, as Pierre Dubois stated in his comment. Perhaps the references mentioned here can be helpful to you: How do you see that higher genus surfaces are not homogeneous?