This article provides an overview of the cubic phase function (CPF) as a tool proposed for both parametric and nonparametric estimation of the frequency modulated (FM) and in particular polynomial phase signals (PPS). This simple tool motivated small revolution in this field with numerous extensions and applications. We are describing the CPF and compare some of its extensions for both one-dimensional and twodimensional signals. The comparisons are performed in terms of accuracy (measured with signal-to-noise (SNR) threshold and mean-squared error (MSE)) and computational complexity. Also, we review the CPF and related transforms applications.