The victims, aged 21, 23 and 33, were in a garden in Gartcraig Road, Carntyne, Glasgow, Police Scotland said.
All three were taken to Glasgow Royal Infirmary after police were called at around 9.35pm on Friday.
The 33-year-old remains in hospital in a serious but stable condition, according to police.
The suspect is white, in his 20s with an athletic build. He has shaved brown hair and was wearing a light grey hooded top with a white zip and three-quarter length white shorts with a black design.
Detective Sergeant John Semple said: “The motive for this attack remains unclear and I am appealing to anyone who may have any information or knowledge to get in touch with us.
“On that particular night, people may have been out and about, perhaps sitting in their gardens, enjoying the good weather and they may have seen or heard something which could assist our enquiries. This can also be a busy road with cars and taxis and if you were in the area around the time of the incident, please do contact us.”