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Robertson, 30, was most recently at AFC Wimbledon, where he made 13 outings for the Dons, scoring one goal.
The Dundee-born defender, who can play in the centre or on the right of defence has also spent time at Sheffield United, Preston North End and Port Vale, among others.
He began his career at the Blades in 2005 before moving on to Torquay United where he made a total of 179 appearances across the National League and League Two before moving to League One Preston in 2011, where he spent two years.
Robertson then transferred to Port Vale, in League One, and was there for a further two years and he made 61 appearances for the Valiants, scoring three goals before a one-year stint at SPL outfit Ross County.
He returned to England in 2016 to sign for Wimbledon, but became a free agent this summer.
Speaking to the Swindon Town website, Robertson said: “I’m delighted to be here.
Swindon Town have not yet revealed the contract length of Robertson.
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