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Brand was to perform in Auckland on today and tomorrow, in Wellington on Monday and Christchurch on Tuesday.
The promoters of the Russell Brand Recovery Live Tour said tickets can be refunded from the point of purchase.
Taking to Twitter, the celebrity said "rapidly evolving challenges and guidelines related to travel and gathering" were to blame.
"I am postponing all tour dates until we better understand coronavirus and how to contend with it. I hope I see you soon," he said.
On Tuesday, a sold-out show in Perth, Australia, was cancelled also over coronavirus fears.
OTHER ACTS CANCEL
Alt metal band Deftones postponed their March 15 show at Auckland's Trusts Arena, as My Chemical Romance announced their March 24 show in Western Springs would also not go ahead.
Electronic musician Marc Rebillet has postponed his Australia and New Zealand tour, as he was due to play at Auckland's Powerstation on March 24.
The World Health Organisation this week declared the virus a pandemic, as the number of cases reached 138,000 globally, with just over 5000 deaths.