Very early launches used monkeys on top of a V2 rocket. It appears the rockets are much better now for launching more advanced primates.
“Another rhesus monkey named Albert II, for example, became the first primate to reach space, achieving an altitude of 83 miles (134 km) aboard another V2 in June 1949. He survived the launch but died after a parachute failure caused his capsule to slam hard into the ground”.