  We introduce the notion of a Seshadri stratification on an embedded
projective variety. Such a structure enables us to construct a Newton-Okounkov
simplicial complex and a flat degeneration of the projective variety into a
union of toric varieties. We show that the Seshadri stratification provides a
geometric setup for a standard monomial theory. In this framework,
Lakshmibai-Seshadri paths for Schubert varieties get a geometric interpretation
as successive vanishing orders of regular functions.
