This article is a revised version of the text of the plenary conference I gave at the XIX Congress of ``Unione Matematica Italiana'', held in Bologna in September 2011. It discusses the arithmetic significance of the values at integers of the complex and p-adic L-functions associated to Dirichlet characters and to elliptic curves.
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Con il contributo del Ministero per i Beni e le Attività Culturali