TITLE: Stable infinity categories vs dg-categories
QUESTION [20 upvotes]: What is the relation between dg-categories and stable $\infty$-categories?
Given a dg-category one can form its dg-nerve and get a $\infty$-category
(which will be stable if the dg-category is?). 
Can one turn a stable $\infty$-category into a dg-category or $A_\infty$-category somehow?
I have heard the statement that at least over a field of characteristic zero the theories
of stable $\infty$-categories and dg-categories are "equivalent".
What would be a precise formulation of this statement and what would be a reference?

REPLY [19 votes]: See the recent paper

Lee Cohn, Differential graded categories are k-linear stable infinity categories, arXiv:1308.2587

where a proof has been written down.  The precise statement is that the underlying $(\infty,1)$-category associated to the Morita model structure on dg-categories over $k$ (where fibrant objects are karoubian pretriangulated dg-categories) is equivalent to the $(\infty,1)$-category of karoubian stable $k$-linear $(\infty,1)$-categories.
Update: Though Cohn works over characteristic zero, Bertrand Toen told me his arguments in fact hold in arbitrary characteristic.
Also, see the new paper 

Giovanni Faonte, Simplicial nerve of an A-infinity category, arXiv:1312.2127

where it is proved that the dg-nerve functor takes pretriangulated dg-categories to stable $(\infty,1)$-categories.