The decision, which was announced this afternoon by the R&A, means the 150th staging will now take place in 2022 at St Andrews.
It will be the first time the world's oldest major hasn't been played since 1945 due to the Second World War.
The event in Kent is the only men's major of the year to be cancelled outright.
The Masters, the US PGA Championship and the US Open have all now been rescheduled for later in the year.
Martin Slumbers, Chief Executive of The R&A, said: “Our absolute priority is to protect the health and safety of the fans, players, officials, volunteers and staff involved in The Open.
"We care deeply about this historic Championship and have made this decision with a heavy heart. We appreciate that this will be disappointing for a great many people around the world but this pandemic is severely affecting the UK and we have to act responsibly. It is the right thing to do.
“I can assure everyone that we have explored every option for playing The Open this year but it is not going to be possible.
“There are many different considerations that go into organising a major sporting event of this scale.
"We rely on the support of the emergency services, local authorities and a range of other organisations to stage the Championship and it would be unreasonable to place any additional demands on them when they have far more urgent priorities to deal with.
"In recent weeks we have been working closely with those organisations as well as Royal St George’s, St Andrews Links Trust and the other golf bodies to resolve the remaining external factors and have done so as soon as we possibly could. We are grateful to all of them for their assistance and co-operation throughout this process.
“Most of all I would like to thank our fans around the world and all of our partners for their support and understanding. At a difficult time like this we have to recognise that sport must stand aside to let people focus on keeping themselves and their families healthy and safe.
"We are committed to supporting our wider community in the weeks and months ahead and will do everything in our power to help golf come through this crisis.”
Reigning Open champion Shane Lowry said: "Obviously I'm disappointed that I won’t get to defend the Open Championship this year.
"But I feel the R&A have made the right decisions based on people’s health and safety. See you all at Royal St George’s in 2021."
The R&A will transfer over tickets and hospitality packages purchased for the Championship in 2020 to The Open in 2021.
Purchasers who no longer wish to (or are no longer able to) attend in 2021 will receive a full refund. Further information on this process will be sent directly to ticket and hospitality purchasers in due course.
The R&A’s remaining professional and amateur championships scheduled this year are under review and updates will be given on any further changes.
These include the AIG Women's British Open, which is due to be played at Royal Troon for the first time in August.
The decision about The Open was quickly followed by the the main golfing bodies and tours unveiling a revised calendar of events for 2020.
The US PGA Championship has been rescheduled for Harding Park in San Francisco in the first week of August.
The US Open at Winged Foot has been moved to 17-20 September - the week before the Ryder Cup, which remains in its original slot at Whistling Straits.
The Masters, meanwhile, has been slotted in for 12-15 November.