A headstone was vandalised at a church graveyard on the Lurgan Road in Banbridge.
Grave ornaments and flowers were thrown around two other gravesites and there was also some evidence of drug misuse in the area, according to police.
Sergeant Stewart said police received a report shortly after 11.30am on Sunday that a headstone had been vandalised and kicked over, causing it to split.
“This is an appalling act of criminal damage. A graveyard is a place for quiet reflection to peacefully pay respects to loved ones who are no longer with us.
“The damage caused to these gravesites shows a total lack of respect and is very upsetting for grieving loved ones.
“Our enquiries into this incident are ongoing. We would ask if you saw anyone acting suspiciously in the area or if you have information that would help our investigation to contact us on the non-emergency number 101 quoting reference number 860 of 29/05/22,” said the sergeant.
You can also submit a report online using the non-emergency reporting form via http://www.psni.police.uk/makeareport/.
Alternatively, information can also be provided to the independent charity Crimestoppers on 0800 555 111 which is 100% anonymous and gives people the power to speak up and stop crime.