New figures from the Central Statistics Office show that the total number of trips to Ireland increased by 10.5% to 3,061,500 in the three months from July to September.
This was an overall increase of 290,600 compared to the same time a year ago and was the best performance on record for the key summer period.
The CSO said that trips here by people in the UK increased by 9.3% to 1,148,500 despite weaker sterling making such a trip more expensive.
Trips by residents of European Countries rose by 10% to 1,085,300, while trips by residents of North America to Ireland increased by 15.9% to 637,500.
The CSO figures also reveal that the total number of overseas trips made by Irish residents during July to September increased by 3% to 2,252,900.
Commenting on today's CSO figures, Niall Gibbons, the CEO of Tourism Ireland, said they indicate that this was the best ever month of September for Irish tourism, with almost 900,000 arrivals recorded.
Mr Gibbons said it is clear already that 2016 will be another record-breaking year for Irish tourism.
"Upcoming promotions - including a significant presence at World Travel Market in London, with about 70 Irish tourism partners; a sales mission to Australia and New Zealand; and our ‘Jump into Ireland’ travel trade and media blitz to the west coast of the US next month - will help to keep this momentum going and deliver further success in 2017 and beyond," he added.