That Stein manifolds have all $(p,q), p \geq 0, q \geq 1$ vanishing Dolbeault cohomology groups is more or less standard. I am a little bit confused about the reverse implication: whether the vanishing of all Dolbeault cohomologies ($p \geq 0, q \geq 1$) implies Steinness? I could not find a reference, yet I noticed the following:


In the recent (and quite authoritative) book by Forstnerič ''Stein Manifolds and Holomorphic Mappings'' Theorem 2.4.6 says:



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> On any Stein manifold $X$ the Dolbeault cohomology groups vanish: $H^{p,q}\_{\bar\partial}(X) = 0$ for all $p \geq 0, q \geq 1$.
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So the reverse is not even mentioned.


In the older book by Grauert and Remmert ''Theory of Stein spaces'' on page 81 one reads:



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> The assumptions of Theorem 5 are always satisfied by Stein manifolds. It is not known if there exist non-Stein manifolds of this type (i.e. $H^q(X,\Omega^p)=0, p \geq 0, q \geq 1$)
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So I assume this was open at least as of 2004 (?)


On the other hand a recent paper by Chakrabarti and Shaw entitled ''The
$L^2$-cohomology of a bounded smooth Stein domain is not necessarily Hausdorff'' (on [arXiv](http://arxiv.org/abs/1305.5924), the author's website says that it is to appear in Math. Ann.) begins with:



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> For each bidegree $(p,q)$, with $p\geq0,q >0$, the Dolbeault Cohomology group $H^{(p,q)}(\Omega)$ of a Stein manifold $\Omega$ in degree $(p,q)$ vanishes, and indeed this property characterizes Stein manifolds among complex manifolds.
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The provide references for that fact are the books of Hörmander and Gunning-Rossi (where I couldn't find it).



Vanishing of Dolbeault cohomologies and Steinness
The reference to the books of Hörmander and Gunning-Rossi only seems to cover a special case of this: a domain $X$ in $\mathbb{C}^n$ is a Stein manifold if and only if $H^i(X,\mathcal{O}\_X)=0$ for all $i$ with $1\leq i\leq n-1$. Another possible reference is Theorem 63.7 in 


* L. and B. Kaup: Holomorphic functions of several variables. De Gruyter, 1983.


A similar result is stated (with similar references to Hörmander and Gunning-Rossi) for open subsets in Stein manifolds in 


* M.G. Eastwood and G. Vigna Suria. Cohomologically complete and pseudoconvex domains. Comment. Math. Helvetici 55 (1980), 413-426.


These results both assume that the complex manifold sits inside a Stein manifold to start with. For an absolute statement, you need the vanishing of cohomology for all coherent analytic sheaves, as mentioned in Peter Dalakov's comment. (However, unless I am missing something, this does not seem obviously equivalent to vanishing of Dolbeault cohomology.)


