Organisers said an 18-year-old Australian from Melbourne and a 20-year-old Spaniard from the northern city of Saragossa were hurt but no details were released about the extent of their injuries.
Later, a 43-year old Irishman was gored through his leg, while a 21 year-old Spaniard was gored in the chest.
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However, neither was believed to have been touched by the bulls' horns which can cause serious injury or even death.
The 825-metre run started at 8:00 am. The nine-day festival ends on July 12.
The bull run spectacle, popularised by Ernest Hemingway's 1926 novel The Sun Also Rises, involves runners trying to outrace six bulls released every day to run from their corral through the old town's narrow, cobbled stone streets.
The bull runs have claimed 15 lives since 1911 and each year dozens of runners are injured.
Last year, a bull gored a 27-year-old Spaniard to death, piercing his neck, heart and lungs with its horns in front of the hordes of tourists.
The mayor of Pamplona has this year banned the noisy South African trumpets called vuvuzelas, which have become internationally popular with the current football World Cup, because of the nuisance he said they posed to neighbours.
The British embassy this year posted a warning telling Britons to seriously consider the risks if they join in the run.
The festival also features a range of concerts, street parties and dances, and the revelers' nightly ritual of spraying each other with red wine.
