Two historic Melbourne monuments were vandalised as tens of thousands of people took to the streets to protest the controversial Australia Day holiday.

A statue of John Batman, a founder of Australia's second largest city who was also involved in the deaths of indigenous people, was sawn in half and a monument commemorating Australian soldiers in World War One was covered in red paint.

Australian correspondent Oliver Peterson says police have launched investigations into these incidents - as well as similar acts of vandalism across the country.

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