The Scottish Government said the move will come into effect on February 11.
It follows a similar announcement in England, with UK transport secretary Grant Shapps insisting the policy of post-arrival lateral flow tests had “outlived its usefulness”.
Testing will no longer be a requirement for all adults who have completed a full course of an approved vaccine, usually at least two doses.
Children under the age of 18 continue to be treated as fully vaccinated.
However, travellers will still need to fill in passenger locator forms and face coverings will still be required at Scottish airports in line with wider health advice.
Those who are not vaccinated will be required to take pre-departure tests and a PCR test on or before day two.
But the requirement for isolation will end and they will no longer have to take a day eight test.
Airport bosses welcomed the move and said they expected to see a “surge in demand as we move ever closer to normality”.
SNP transport secretary Michael Matheson said: “These measures will significantly open up international travel and were agreed on a UK wide basis.
“The measures will be extremely welcome for the Scottish tourism and aviation sectors, encouraging travel from our airports.
“While this is a positive step which will be welcomed by many, we believe further surveillance measures will be necessary across all nations – as intelligence will help in terms of variants of concern.
"It was agreed further work to take this forward will be carried out over the coming weeks.”
Gordon Dewar, chief executive of Edinburgh Airport, said: "We are glad that our governments have realised the time is right to make these changes and give this industry and others that rely on it the best possible chance to stage a recovery.
"The pent-up demand for travel is there as people look to reconnect with family and friends, take that long-awaited holiday, finally go on honeymoon or take that important business trip.
"The industry has long planned for the resumption of safe travel and although the changes will not come into force for a few weeks yet, we do expect a surge in demand as we move ever closer to normality.
"It will be challenging as the industry landscape has changed and we encourage governments to work with the industry to devise a number of initiatives to recover lost connectivity and promote Scotland as a great destination to visit, and we look forward to discussing this with them.”
Kate Nicholls, chief executive of UKHospitality, described the move as a welcome “shot in the arm”.
She said: “This is a sensible and pragmatic step towards normality, given the encouraging fall in cases, the general easing of restrictions and given the UK’s position as a leading destination for international tourists and as a major business and commerce hub.
“This is a very welcome shot in the arm for hotels and tourism-reliant businesses, as we learn to live with the virus.
"It will deliver much-needed confidence for international travellers destined for the UK.
"Our hospitality and tourism industries have endured extremely challenging times and our hope is that this marks the start of a more benign phase for both society and business.
“It comes at the start of what can be a very positive year for the UK economy and for tourism particularly, given the calendar ahead which includes the Queen’s Platinum Jubilee and the Commonwealth Games.
"With the right conditions and support, this industry can help to drive the recovery, to growth and job creation, and reassert the UK’s position as a leading destination.
"To support this year of revival we would urge the Government to extend the reduced rate of VAT for hospitality and tourism.”
Mr Shapps told the Commons the change would help the aviation industry “take back to the skies”.
He said: “We promised we wouldn’t keep these measures in place a day longer than was necessary and it’s obvious to me now that border testing for vaccinated travellers has outlived its usefulness, and we’re therefore scrapping all travel tests for vaccinated people.”
Mr Shapps insisted this would help “ensure that 2022 is the year in which restrictions on travel, on lockdowns and limits on people’s lives are firmly placed in the past”.
EasyJet chief executive Johan Lundgren welcomed the announcement and said: “We believe testing for travel should now firmly become a thing of the past.
“It is clear travel restrictions did not materially slow the spread of Omicron in the UK and so it is important that there are no more knee-jerk reactions to future variants.”
Tim Alderslade, chief executive of airline industry body Airlines UK, said the announcement was “a landmark day for passengers, businesses and UK plc”.
The update around testing comes after Easyjet told The Scotsman it had already seen signs people plan to take two holidays this year “to make up for lost time” after Covid halted much foreign travel over the past two summers.
Scotland’s biggest airline has also forecast the impending easing of work from home restrictions in Scotland will have a “huge impact” on UK air travel such as business trips to London.
Easyjet UK country manager Ali Gayward had told The Scotsman: "We're waiting with bated breath to see quite what the response is for the relaxation of work from home restrictions.
"We would see an awful lot of regular commuters travel from Scotland to London every week, or down to the south west or even over to Northern Ireland.
"A large proportion of our flying from Scotland to London is those weekly commuters, who typically tend to live in Scotland, but work in places like Canary Wharf, such as in financial services.
"Banks and financial services are asking their employees to come back now."