Two-pion production reactions in proton–proton collisions have been studied using the PROMICE/ WASA detector and an internal cluster gas-jet target at the CELSIUS storage ring in Uppsala. The total cross sections for the pp→ppπ +π − , pp→ppπ 0π 0 and the pp→pnπ 0π + reactions are presented at beam energies ranging from 650 to 775 MeV. An isospin analysis of the amplitudes involved in these three reactions indicates that the excitation of the N ∗(1440) followed by direct two pion emission can be the most important amplitude when allowed. However, large contributions from other production mechanisms must also be prevalent to explain the large pp→pnπ 0π + cross section. The data are compared to an extensive model including non-resonant as well as resonant production via N ∗ and Δ excitation.