In this paper, we first describe the paleointensity furnaces and the procedures specifically designed to minimize oxidation of the samples during Thellier and Thellier paleointensity experiments, routinely used at Gif. We also propose some improvements to the set of criteria commonly used to select the results of the Thellier experiments, in order to further increase the reliability of paleointensity determination finally to be included in databases. The new set of paleointensity criteria (PICRIT-03 = PaleoIntensity CRIteria in 2003) is discussed and the results obtained from two long Hawaiian basaltic cores, the Scientific Observation Hole-4 (SOH-4) and the Hawaiian Scientific Drilling Project-1 (HSDP1) are used to illustrate its effectiveness. Some of the results are re-evaluated and a new composite record of absolute paleointensity and inclination spanning 420kyr is presented. The inclination values obtained from the two cores combined support the hypothesis of persistent shallow inclinations at Hawaii. In intensity, the upper 100kyr are covered by the data from SOH-4, with 309 reliable determinations (out of 766 examined) obtained from 143units which belong to the Kilauea volcano at the top of this core. The 341 reliable data covering the 100–420kyr interval are from 149units of core HSDP1, belonging to the Mauna Kea volcano. The interval 320 to 420kyr is particularly detailed, because of high extrusion rate of the volcano. The use of more stringent criteria decreases the number of determinations considered as reliable in previous publications but it improves their accuracy and robustness.