During the incident on the evening of March 1, a man gained access to the museum and a display cabinet was forced open and the five medals were taken.
The five medals had been awarded to local engineering firm Davey Paxman & Co in the late 19th century.
Those stolen were bronze medals from the Paris Exhibition 1878 and from the Sydney International Exhibition 1879 plus gold medals from the London International Exhibition of Mining and Metallurgy 1890, the Crystal Palace International Exhibition 1882 and the Crystal Palace Electrical Exhibition in 1892.
Darius Laws, Colchester Borough Council portfolio holder for Economy, Business & Heritage, said: “I would also appeal to the person who has taken the medals, or now has the medals, to now arrange for their safe return either to the museum or to the castle. More than any financial loss, this is a loss to Colchester’s heritage and it is that loss which can never be replaced."
The police have asked anyone with information about the incident to get in touch with Essex police by calling 101 quoting crime reference number 42/53042/22 or contact Crimestoppers anonymously on 0800 555 111.
To view CCTV images of the man who broke in, visit the Essex police website.