They are the most extraordinary of fish, resembling "hedge trimmers with fins".
The sawfish, which is a kind of ray, is also among the most endangered of the fish living in the oceans.
Once found along the coastlines of 90 countries, the animals are now presumed extinct in more than half of these, according to a new study.
They are vanishing due to habitat loss and entanglement in fishing nets, experts have said.
Their "saws", which evolved to sense and attack prey, have now become a liability, making them prone to being caught up in fishing gear.
"Through the plight of sawfish, we are documenting the first cases of a wide-ranging marine fish being driven to local extinction by overfishing," said Prof Nick Dulvy of Simon Fraser University (SFU) in British Columbia, Canada.
Of the five species of sawfish, three are critically endangered, while two are listed as endangered, according to the International Union for Conservation of Nature's (IUCN) Red List of Threatened Species.
Previously widespread, the sawfish are now presumed extinct in 55 nations, the study said.
There are 18 countries where at least one species of sawfish is missing, and 28 more where two species have disappeared.
The list of countries where sawfish are extinct now includes China, Iraq, Haiti, Japan, Timor-Leste, El Salvador, Taiwan, Djibouti and Brunei.
The US and Australia appear to be the last strongholds for the species, regarded as "lifeboat nations," where sawfish are better protected.
The study, published in Science Advances, also identified eight nations where urgent action could make a big contribution to saving the species through conservation efforts.
These are Cuba, Tanzania, Colombia, Madagascar, Panama, Brazil, Mexico and Sri Lanka.
"While the situation is dire, we hope to offset the bad news by highlighting our informed identification of these priority nations with hope for saving sawfish in their waters," said Helen Yan of SFU.
She said it is still possible to restore sawfish to more than 70% of their historical range, "if we act now".
International trade in sawfish is prohibited under the Convention on International Trade in Endangered Species of Flora and Fauna, but targeted and accidental killings still occur.
Sawfish fins and teeth are sold as trophies, food or medicine, and as spurs for cockfighting.
Sonja Fordham, a researcher of the study and president of Shark Advocates International, said there were opportunities to "bring these extraordinary animals back from the brink".
But she warned that in too many places, "we're running out of time to save them".
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