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The app launched on March 17th, 2016 for Japan, and on March 31st, 2016 for other parts of the world. On April 27, 2016 Miitomo was announced to have over 10 Million users across iOS and Android. On June 9, 2016, Miitomo was announced for Mexico and Brazil, officially releasing in those countries on June 30, 2016 and July 28, 2016 respectively.
As Miitomo has officially been discontinued, the below downloads are kept as reference. The app is still downloadable, but will never connect to a server.
For the official website of Miitomo, it can be viewed on the Internet Archive Wayback Machine. Connecting to the official website today will redirect to the Nintendo official website.