  Riemann sums, a classical method for approximating the definite integral of a
function, have been extensively studied in the past. However, their monotonic
properties, while still of great importance, particularly in approximation
theory and interpolation theory, remain somewhat obscure. This paper is
dedicated to proving general theorems about the monotonicity of left and right
Riemann sums, a problem first raised by Fej\'er in 1950. We provide a
much-needed review of the literature on the problem and offer several new
sufficient and necessary conditions for the monotonicity of Riemann sums.
Additionally, we present a new insightful proof of a fundamental theorem
related to these sums using tools from the theory of majorization. The author
also delves deeper into a question posed by Borwein, almost resolving it
completely.
