  The notions of $\mathbb Q$-Gorenstein scheme and of $\mathbb Q$-Gorenstein
morphism are introduced for locally Noetherian schemes by dualizing complexes
and (relative) canonical sheaves. These cover all the previously known notions
of $\mathbb Q$-Gorenstein algebraic variety and of $\mathbb Q$-Gorenstein
deformation satisfying Koll\'ar condition, over a field. By studies on relative
$\mathbf S_{2}$-condition and base change properties, valuable results are
proved for $\mathbb Q$-Gorenstein morphisms, which include infinitesimal
criterion, valuative criterion, $\mathbb Q$-Gorenstein refinement, and so
forth.
