In Spain, where parts of the country will be in the path of totality, finding certified protective glasses has become increasingly difficult, with many distribution points reporting they have run out.
A notice posted at the entrance of Madrid's National Museum of Natural Sciences said its supply of protective glasses was exhausted.
The museum was among 30 science institutions across the country taking part in a campaign organised by the government to distribute two million pairs of glasses for free.
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Some pharmacies in central Madrid still had supplies Tuesday, generally selling certified glasses for safely viewing the Sun for between €3-€4 ($4.60) a pair.
But Carles Sisquella, who runs an optical shop in Barcelona, said his store had been out of stock for 10 days.
"There are now a lot of people who obviously won't be able to get glasses," he said.
Doctors warn that looking directly at the Sun during a solar eclipse can damage the retina, potentially harming the cells responsible for sharp central vision.
"I've been asking around since Friday, but they're nowhere to be found," a woman named Mabel told AFP in Barcelona after another fruitless search.
READ ALSO: How to get free glasses to see Spain's total solar eclipse
'Gone in 30 minutes'
The rush is being repeated across Europe.
In Brussels, where about 90 percent of the Sun will be covered by the Moon, people were also racing to find glasses.
With a seeming majority of opticians and pharmacists out of stock, pastry chef Arno Dossogne, 37, struck gold at a small opticians' store that received a last-minute shipment Monday afternoon.
"I got some for me, my boyfriend and a couple of friends whose online order got cancelled!" he told AFP.
"It was this or nothing. You can't mess around, or you'll burn your eyes."
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The opticians' manager Fouad Mehdaoui told AFP he acted on a tip from his pharmacist wife, placing large orders of glasses early on, but even then was outrun by demand and had to order more.
"Two boxes just came in with 100 pairs in total -- they were gone in 30 minutes," he told AFP as a stream of eclipse enthusiasts filed in the store, by chance or word of mouth, and an employee fielded non-stop phone enquiries.
In France, where the eclipse will be partial, some pharmacies have been stripped of their supplies while others were still receiving shipments -- and charging significantly more.
Customers lined up outside La Pharmacie du Soleil in Paris on Monday afternoon to buy glasses for €5.95 a pair, only to be disappointed when staff displayed an empty box.
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'Pure madness'
The pharmacist asked customers to check back later, saying another shipment of 500 pairs was expected that evening.
"Come back in an hour or tomorrow morning," he told customers. "It's pure madness."
A pharmacy employee said the supplier had been so overwhelmed by orders that it could no longer make deliveries itself, forcing it to ship the glasses by Uber.
Philippe Besset, president of France's Federation of Pharmaceutical Trade Unions, said demand had been far greater than expected.
A few months ago, suppliers had offered pharmacies around 200 pairs each, enough to equip about four million people, he told AFP.
"But obviously more people want to watch the eclipse," he said.
His pharmacy sold its entire order of 200 pairs within a week.
In Portugal, the Club da Visao network of more than 600 opticians had been offering certified glasses free of charge, limited to one pair per person.
Despite several restocks, supplies had run out at many locations.
The shortage has prompted eye specialists to warn people against buying uncertified alternatives.
In Spain, authorities on Monday withdrew six models from the market because of the risk of eye injuries.
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