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?