"They have splashed spurious claims with bombastic titles on their front pages", Rush said.
Rush said his wife, daughter and son had also been hurt by the allegations.
The veteran Australian star launched action in the Federal Court against the Murdoch-owned publication to "redress the slurs, innuendo and hyperbole that they have created around my standing in the entertainment industry and in the greater community".
It comes after the Oscar victor stood down from his role as president of the Australian Academy of Cinema and Television Arts in light of the allegations.
The AACTA said in a statement posted on its website on December 2 that the organization "acknowledges the decision today of Geoffrey Rush to voluntarily step aside as President of AACTA and accepts and respects his decision to do so". Though the exact nature of the complaint remains unknown, the Telegraph claimed the allegation was so serious that the Sydney Theatre Company would never work with the actor again.
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The legal team pointed out that The Daily Telegraph published in print the allegations about their famous client alongside an article "about allegations concerning alleged sexual predator and television personality Don Burke so as to falsely and unfairly associate the allegations against the applicant with the allegations against Mr Burke".
In November the Sydney Theatre Company released a statement to the Daily Telegraph saying it had received a complaint against Rush from someone who accused him of "inappropriate behaviour" during the company's staging of William Shakespeare's King Lear two years ago.
"This has created irreparable damage to my reputation", he said.
The Daily Telegraph editor Chris Dore said they would defend the newspaper's reporting in court.
The matter is expected to reach court early in 2018.