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Given a finite morphism of smooth curves, the Prym variety parametrizes line bundles on the source whose norm is isomorphic to the canonical bundle of the target. What if the target and/or the source are not smooth? Several people already investigated this question. In this talk I would like to explain how log geometry can be used to tackle this problem, and possibly also outline some applications, e.g. how this machinery can be used to produce (several) compactifications of the the so called universal Prym variety. Joint work with A. Gross, J. Horn, S. Molcho and M. Ulirsch.