The illicit trade may have occurred until as recently as the 1990s.
Women who have been told their babies died, but are convinced they were stolen, are giving DNA samples for a national database, hoping to find out if their suspicions are true.
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Hundreds of women across Spain are coming forward with similar stories.
After Spain's civil war, children were removed from left-wing Republican prisoners and given to supporter's of General Francisco Franco's dictatorship.
Up to 30 000 children from left-wing families were affected, historians estimate.
Some believe this practice later became an illicit business.
Antonio Barroso, who found out his mother bought him as a baby from a Catholic nun, said he is trying to find out where he came from.
"I want to know the truth ... I want to know what happened and who was responsible. And if people need to be punished for that, they should be punished," he told the BBC.
