European companies are promising billions in new deals in Iran—€5.7 billion from Italian steelmaker Danieli and more than €300 million from French car builder PSA Peugeot Citroën, among others—as Iranian President Hassan Rouhani works to revive trade and political ties on the Continent this week.

Mr. Rouhani landed in Rome on Monday on his first overseas trip since the European Union lifted sanctions on Jan. 16 in return for Tehran’s implementation of key restrictions on its nuclear program.

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