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

Rota's program on algebraic operators, rewriting systems and Gr\"obner-Shirshov bases