In his inaugural speech, he said ending the conflict with Russian-backed rebels in the east would be his top priority, the BBC reported.
Zelensky, who has no previous political experience, scored a landslide victory in last month's presidential election.
"We must become Icelanders in football, Israelis in defending our land, Japanese in technology," he said.
Hinting at Russia, without naming it, he said Ukrain must also "become Swiss in our ability to live happily with each other, despite any differences".
"Our first task is to achieve a ceasefire in Donbass," he said, referring to the eastern region controlled by Russian-backed rebels.
In the election, the former television actor ousted incumbent Petro Poroshenko, who had been in power since 2014.
Zelensky has promised to tackle corruption, but has given few details of his plans since winning by a landslide on April 21.
At the ceremony, Zelensky was given golden symbols of office, including a sceptre, which he held aloft in a victory salute. A choir in folk costume sang patriotic hymns to welcome him.
Russian state TV said no Russian official had been invited to the inauguration.
Referring to the conflict in the east, Zelensky switched to Russian, saying: "I'm convinced that for this dialogue to start, we must see the return of all Ukrainian prisoners" - apparently a message to the Russian government.
There has been much political wrangling over the timing of Zelensky's inauguration. He had wanted to hold it on Sunday, a day of mourning for victims of Stalin-era repression in the country.
But parliament disagreed and voted to hold it on Monday morning.