  An exponential polynomial is a finite linear sum of terms $P(z)e^{Q(z)}$,
where $P(z)$ and $Q(z)$ are polynomials. The early results on the value
distribution of exponential polynomials can be traced back to Georg P\'olya's
paper published in 1920, while the latest results have come out in 2021.
Despite of over a century of research work, many intriguing problems on value
distribution of exponential polynomials still remain unsolved. The role of
exponential polynomials and their quotients in the theories of
linear/non-linear differential equations, oscillation theory and
differential-difference equations will also be discussed. Thirteen open
problems are given to motivate the readers for further research in these
topics.
