Cricket Australia said that advice from the country's security and intelligence organisation and department of foreign affairs about an increased risk for the team meant this month's matches could not proceed.
CA chief executive James Sutherland said an independent security assessment confirmed the "risk of terrorism in Bangladesh targeting Australian nationals".
"We have decided that, regrettably, we have no alternative but to postpone the tour," Sutherland said.
Australia had been scheduled to depart Sunday for the first Test starting October 9 in Chittagong.
Also Read
Nazmul Hassan, the Bangladesh Cricket Board (BCB) president, expressed sadness at the decision, insisting that measures to tighten security should have satisfied any safety concerns.
"Many countries face this kind of terrorist threat but it never stops cricket," he told reporters in Dhaka.
"With the kind of blanket security that we had offered them, I can't see how they would have faced any problems."
"We had hoped that the security concerns would fade, but unfortunately the advice we have received from government, our own security experts and independent security advisors has clearly indicated that there are now high risks to our people should they make the trip," Sutherland said.
"We have worked tirelessly to try to find a way for the tour to proceed, but in the end it was simply not possible," he said.