We will discuss the importance of metrics in spectral analysis. We will motivate and compare two natural alternatives with substantially different properties. The first quantifies differences between two random processes in the context of a prediction problem, and because of that, it turns out be ``blind'' to the presence of deterministic components in power spectra. The second metric is designed to satisfy a ``wish list'' of desirable properties and relies on ideas in the Monge-Kantorovich transportation literature. The relevance of the two metrics will be contrasted in selected applications.