In this talk, we shall first see some examples of a minimal graded free

resolution of a finitely generated graded module $M$ over a commutative ring $R$. Given a

field $K$, a positively graded $K$-algebra $R=\bigoplus_{i \in \mathbb{N}} R_i$

with $R_0=K$ is \emph{Koszul} if the field $K$ has an $R$-linear free resolution when viewed as

an $R$-module via the identification $K=R/R_{+}$.

We shall review the classical invariant Castelnouvo-Mumford regularity of a module and define

Koszul algebras in terms of regularity. 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. At the end, I will present

recent progress on the Charney-Davis-Stanley conjecture and on several problems concerning Koszul algebras.