  We introduce a notion of Homological Projective Duality for smooth algebraic
varieties in dual projective spaces, a homological extension of the classical
projective duality. If algebraic varieties $X$ and $Y$ in dual projective
spaces are Homologically Projectively Dual, then we prove that the orthogonal
linear sections of $X$ and $Y$ admit semiorthogonal decompositions with an
equivalent nontrivial component. In particular, it follows that triangulated
categories of singularities of these sections are equivalent. We also
investigate Homological Projective Duality for projectivizations of vector
bundles.
