Between 15 and 20 gravestones at the St Martin’s Cemetery in Whittingehame Drive have been pushed over within the past two weeks, resulting in many being broken.
The entrance gate to the site has also been damaged and the estimated cost of this destruction around £6,500.
Officers are now keen to hear from anyone who can assist with their enquiries into these incidents.
Inspector Bob Rodriguez said: “This is an abhorrent crime that has caused genuine distress to the families and friends of those, whose graves were affected.
“Anyone who remembers seeing any suspicious activity around the cemetery in the past couple of weeks is asked to contact police immediately.
“Similarly, we would urge anyone with information that can help trace those responsible to also come forward.”
An East Lothian Council spokesperson added: “We are appalled by the callousness involved in these acts of vandalism and the distress they have caused relatives and others visiting St Martin’s cemetery.
“We hope that anyone who knows anything will come forward and help the police track the vandals down.