  We prove a version of Koszul duality and the induced derived equivalence for
Adams connected $A_\infty$-algebras that generalizes the classical
Beilinson-Ginzburg-Soergel Koszul duality. As an immediate consequence, we give
a version of the Bern\v{s}te{\u\i}n-Gel'fand-Gel'fand correspondence for Adams
connected $A_\infty$-algebras.
  We give various applications. For example, a connected graded algebra $A$ is
Artin-Schelter regular if and only if its Ext-algebra $\Ext^\ast_A(k,k)$ is
Frobenius. This generalizes a result of Smith in the Koszul case. If $A$ is
Koszul and if both $A$ and its Koszul dual $A^!$ are noetherian satisfying a
polynomial identity, then $A$ is Gorenstein if and only if $A^!$ is. The last
statement implies that a certain Calabi-Yau property is preserved under Koszul
duality.
