Koszul algebras are the algebras over which the resolution of the
residue class field is given entirely by linear matrices. In this talk, we
shall first
review the classical invariant Castelnouvo-Mumford regularity of a
finitely generated
graded module $M$ over a commutative ring $R$ and define Koszul algebras
in terms of
regularity. I will give some examples to understand Castelnouvo-Mumford
regularity
and Koszul algebras. We shall also discuss several other characterizations of
Koszul algebras. Then I will present some results on Koszul property of
diagonal subalgebras of bigraded
algebras; in particular, Koszul property of diagonal subalgebras of Rees
algebras for a
complete intersection ideal generated by homogeneous forms of equal degrees.