Investigators from the country's Ministry of Internal Affairs raided the data centre that was hosting the website's servers.
Torrents allow users to download music, video and other internet content by downloading small bits of files from others' computers at the same time.
The shutdown is the latest news in a campaign against file-sharing sites.
It follows the US's closure of Megaupload, and several European ISPs (internet service providers) being ordered to block access to The Pirate Bay.
Demonoid was listed alongside both of these sites in The Notorious Markets List - a document drawn up by the US government at the end of last year highlighting services that "merit further investigation for possible intellectual property rights infringements".
It noted that Demonoid "recently ranked among the top 600 websites in global traffic and the top 300 in US traffic".
Back online?
Users first became aware of the action on 26 July, when attempts to access Demonoid's site yielded a "server busy" message.
The Torrentfreak news site reported that Ukraine's Division of Economic Crimes acted after receiving a request from the international police organisation Interpol.
It said the local authorities then contacted Demonoid's ISP, Colocall, which decided to pull its service, and allowed investigators to copy data off its servers.
"Demonoid is known for its links to relatively rare content which may be harder to come by now," Torrentfreak's editor Ernesto Van Der Sar told the BBC.
"However, it's not going to stop the majority of people from sharing files as the most popular items are available though hundreds of other BitTorrent sites."
The action follows the arrest of one of Demonoid's administrators in Mexico last October. But despite the setbacks Mr Van Der Sar suggested it was too soon to consign the site to history.
"In 2006 The Pirate Bay came back online three days after it was raided, and in the years that followed it grew out to become the largest BitTorrent site," he said.
The BPI, which represents the UK music industry, and the MPAA (Motion Picture Association of America) - which have both campaigned against online copyright infringement - declined to comment when approached by the BBC.
This is kinda old news now, but I've been on vacation.
I can't imagine Demonoid will ever be gone indefinitely. Even if it takes months or years, hopefully it will come back again.Any insight/discussion?
