Paul Johnson was last seen at around 10.45pm on Wednesday, September 15, at Hepburn Gardens, St Andrews, after leaving a property on Hay Fleming Avenue at around 10pm.
The 74-year-old is known to go for long walks in local areas such as the Balgove Golf Course, Craigtoun Country Park and Strathkinness.
Police Scotland has carried out extensive searches across those areas with help from search and rescue partners, police dogs, and the Air Support Unit.
This evening, Wednesday, September 22, officers are revisiting the area where Mr Johnson was last seen in an effort to piece together his last known movements.
In a statement released through Police Scotland, Mr Johnson’s family have said: “We are all terribly worried about Paul and just want him to come home safe.
"We would like to thank the police, the community and all other agencies who have assisted in the search thus far, and we continue to appeal to anyone who believes they may have some information to please come forward and report it to police.”
Mr Johnson has been described as being around 5ft 9in tall and of slim build with grey hair. He was seen wearing a blue rain jacket and light coloured walking trousers.
Inspector Stuart Hay, said: “It has been a week since Paul went missing and during this time extensive areas have been searched by police search teams, partner agencies, and the local community. We are thankful for the support of local residents and Paul’s family who are being kept fully up to date with this enquiry.
“I’d ask that people note Paul’s description and what he is believed to be wearing. Did you see a man matching this description late on Wednesday evening in the St Andrews area, or have you seen him since?
"He is known to go for walks however the time of night he left is out of character and as the passage of time increases we are becoming increasingly concerned for his welfare.”
If you have any information you should contact Police Scotland on 101 quoting incident number 0266 of September 16.