  Many years ago, Rota proposed a program on determining algebraic identities
that can be satisfied by linear operators. After an extended period of dormant,
progress on this program picked up speed in recent years, thanks to
perspectives from operated algebras and Gr\"obner-Shirshov bases. These
advances were achieved in a series of papers from special cases to more general
situations. These perspectives also indicate that Rota's insight can be
manifested very broadly, for other algebraic structures such as Lie algebras,
and further in the context of operads. This paper gives a survey on the
motivation, early developments and recent advances on Rota's program, for
linear operators on associative algebras and Lie algebras. Emphasis will be
given to the applications of rewriting systems and Gr\"obner-Shirshov bases.
Problems, old and new, are proposed throughout the paper to prompt further
developments on Rota's program.
