MOSCOW — A stream of elegant visitors stopped in their tracks on Thursday as they toured Moscow’s new Jewish Museum and Tolerance Center, a sprawling, state-of-the-art complex underwritten by oligarchs close to President Vladimir V. Putin. They had never seen a shtetl like this one.

Touch the screen in one exhibit in this vast building and a visitor can appear in a mirror dressed in the garb of a 19th-century blacksmith, or a trader or a “representative of the intelligentsia.” Tap a Torah in a virtual synagogue, and a cantor’s voice rings in the air. In a virtual Odessa, one can sit down in an interactive cafe to chat with long-dead writers.

Mr. Putin has extended his personal support to the lavish project, donating a month’s salary for its construction, which cost around $50 million. In part because of its scale — organizers say it is the largest Jewish history museum in the world — the project is meant to convey a powerful message to Jews whose ancestors fled or emigrated: Russia wants you back.

President Shimon Peres of Israel, who attended the opening, said it affected him deeply.

“My mother sang to me in Russian, and at the entrance to this museum, memories of my childhood flooded through my mind, and my mother’s voice played in my heart,” said Mr. Peres, 89, who was born in what is now Belarus. “I came here to say thank you. Thank you for a thousand years of hospitality.”