Summary: We study the Wasserstein metric \(W_p\), a notion of distance between two probability distributions, from the perspective of Fourier Analysis and discuss applications. In particular, we bound the Earth Mover Distance \(W_1\) between the distribution of quadratic residues in a finite field \(\mathbb{F}_p\) and uniform distribution by \(\lesssim p^{-1/2}\) (the Polya-Vinogradov inequality implies \(\lesssim p^{-1/2} \log{p})\). We also show that for continuous \(f : \mathbb{T} \rightarrow \mathbb{R}\) with mean value 0 \[ (\text{number of roots of } f) \cdot \left(\sum\limits_{k=1}^{\infty} \frac{|\widehat{f}(k)|^2}{k^2}\right)^{\frac{1}{2}} \gtrsim \frac{\Vert f\Vert^2_{L^1(\mathbb{T})}}{\Vert f\Vert_{L^\infty(\mathbb{T})}}. \] Moreover, we show that for a Laplacian eigenfunction \(-\Delta_g \phi_{\lambda} = \lambda \phi_{\lambda}\) on a compact Riemannian manifold \(W_p (\max \{\phi_\lambda, 0\} dx, \max \{-\phi_{\lambda}, 0\} dx) \lesssim_p \sqrt{\log\lambda/\lambda} \Vert\phi_\lambda\Vert_{L^1}^{1/p}\), which is at most a factor \(\sqrt{\log\lambda}\) away from sharp. Several other problems are discussed.