The Filter Diagonalization Method (FDM) is an elegant and numerically efficient algorithm of spectral estimation. It shows advantages over conventional Fourier transform spectra when the signal-to-noise ratio is adequate, and the true spectrum would be well resolved. Isolated narrow resonances can be captured quite accurately using FDM, even if the time-domain sampling is insufficient to give anything like the true line width by FT. As a true multi-variate method it becomes especially advantageous when applied to multi-dimensional spectra. We hope this chapter will offer a glimpse into the math and science behind Filter Diagonalization, and make the method more ubiquitous.