Two businessmen who want Google to stop linking their names to media reports about past crimes are waiting to see if they have won High Court fights.
Both say their convictions are legally “spent” and argue that they have a right to be forgotten.
Google bosses disagree.
A judge has analysed issues at two High Court trials in London.
Mr Justice Warby finished hearing evidence in the first trial last week, and finished hearing evidence in the second trial on Wednesday.
He is expected to publish a ruling in the near future.
Lawyers say the cases are the first of their kind to be aired in England.
The judge has made orders barring the men from being identified in media reports about the litigation.