Two of the victims, aged 50 and 56, were killed after being violently hit by bulls in the suburbs of the city of Valencia.
Emergency services also confirmed that a French tourist in their 60s died in hospital.
Traditional bull runs are held every summer in Spain as villages celebrate their patron saint.
The most famous bull runs take place as part of the San Fermin festival in Pamplona, where hundreds of runners are chased by six fighting bulls in an 850-metre race through the city's cobbled streets.
This year six people were gored during the festivities, but no serious injuries had been reported.