Most will be installed in rural areas and in new housing developments.
The move comes despite Royal Mail admitting that many post boxes do not cover their costs, as people increasingly use electronic forms of communication.
But the firm said it was part of the first phase of a drive to improve public access to post boxes across the country.
There are around 115,300 post boxes in the UK. Royal Mail said the new installation programme would ensure that more than 98% of households will be within half a mile of a post box.
Plans to install 280 additional post boxes in England, 46 in Wales and 280 in Northern Ireland have also been announced.
The first tranche of installations should happen in spring 2017.
Royal Mail's operations Rob Jenson said: "Access to post boxes is very important to our customers.
"This programme underlines our commitment to them and the Universal Service.
"The new boxes will benefit rural areas, which have not previously been served post boxes.
"We are also installing boxes in urban areas in response to changes in population and in new housing developments throughout the UK."
Rather than decommission uneconomic post boxes, Royal Mail says it has made collections more efficient.
Royal Mail has unveiled plans for about 150 new post boxes in Scotland.