Recently Seshadri stratifications on an embedded projective variety have been introduced by R. Chirivì, X. Fang and P. Littelmann. A Seshadri stratification of an embedded projective variety \(X\) is the datum of a suitable collection of subvarieties \(X_tau\) that are smooth in codimension one, and a collection of suitable homogeneous functions \(f_tau\) on \(X\) indexed by the same finite set. With such a structure, one can construct a Newton-Okounkov simplicial complex and a flat degeneration of the projective variety into a union of toric varieties. Moreover the theory of Seshadri stratifications provides a geometric setup for a standard monomial theory. In the talk, I will introduce the theory of Seshadri stratification and I will give a Seshadri stratification for matrix Schubert varieties, namely varieties of matrices defined by conditions on the rank of some their submatrices.