The signal decomposition of nonlinear and nonstationary data sets is, in general, a challenging inverse problem. Standard techniques, like short time Fourier transform and wavelet transform, are limited in addressing the problem. An alternative way to tackle the problem is to iteratively decompose the signal into simpler components in a data-adaptive way. This is the idea behind what is called the Empirical Mode Decomposition (EMD) method, published originally in 1998. EMD has had and is still having a big impact on many research fields. However, the mathematical properties of EMD and its generalizations, like the Ensemble EMD, are still under investigation. For this reason, an alternative technique, called Fast Iterative Filtering (FIF), was proposed a few years ago [1]. In this talk, we review the mathematical properties of FIF and the decomposition it produces. In particular, we will talk about the energy preservation and orthogonality of this decomposition. Furthermore, we introduce two alternative direct versions of this method called direct FIF (dFIF) and hard thresholding FIF (htFIF). We will show their robustness to noise and ability to produce, in a fast manner, signal decompositions [2]. We conclude by presenting applications of this approach to real-life signals and problems, which are left open in this direction of research. [1] A. Cicone, H. Zhou. Numerical Analysis for Iterative Filtering with New Efficient Implementations Based on FFT. Numerische Mathematik, 147 (1), pages 1-28, 2021. [2] A. Cicone. Iterative Filtering as a direct method for the decomposition of nonstationary signals. Numerical Algorithms, Volume 85, Issue 3, Pages 811-827, 2020.