North Yorkshire Police said the two men got into an "altercation" in the rear smoking area of the Opera House Casino on St Thomas Street at around 5.30am on Saturday December 18.
The men, aged 50 and 34, and a 19-year-old woman, all from Scarborough, required urgent hospital treatment due to their injuries.
The 50-year-old man suffered a "serious" cut to his face, and the 34-year-old required surgery for a "significant" ear injury.
The woman, who was not involved in the incident, suffered a cut to her arm whilst standing nearby.
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Officers are urging a mixed group of bystanders who were near the incident immediately before the alleged assaults, to come forward.
It is believed some of the group were involved in a "heated argument" that may have sparked the violent outburst.
Anyone with information that could assist the investigation should contact North Yorkshire Police on 101, select option 2, and ask for Paul Thompson or Steve Monty at Scarborough CID.
Information can be passed on anonymously to Crimestoppers on 0800 555 111. Quote police reference number 12210263068 when passing on information.