Context. The EUROpean Near Earth Asteroid Research (EURONEAR) is a network which envisions to bring some European contributions into the general context traced by the Spaceguard Foundation which was carried out during the last 15 years mainly by the US with some modest European and amateur contributions. Aims: The number of known near Earth asteroids (NEAs) and potentially hazardous asteroids (PHAs) has increased tremendously, mainly thanks to five major surveys all focused on the discovery of new bodies. But also other facilities are required to follow-up and improvement the orbital parameters and to study the physical properties of the known bodies. These goals are better achieved by a co-ordinated network such as EURONEAR. Methods: Astrometry is mandatory in order to acquire the positional information necessary to define and improve orbits of NEAs and PHAs and to study their trajectories through the solar system, especially in the vicinity of Earth. Photometry is required to derive some physical information about NEAs and PHAs. In order to achieve these objectives, the main method of research of the EURONEAR is the follow-up programme of objects selected by a few criteria, carried out mostly at 1 m-class telescopes endowed with medium and large field cameras. Results: 162 NEAs summing more than 1500 individual positions were observed for a total time of 55 nights in both visiting mode and regular runs using nine telescopes located in four countries. The observations were reduced promptly and reported to the Minor Planet Centre (MPC) which validated and included them in the MPC and NEODyS databases following the improvement of their orbital elements. For one binary NEA we acquired photometry and were able to determine its orbital and rotational periods. Complementary to the follow-up work, as many as 500 unknown moving objects consistent with new Main Belt asteroids and one possible NEA were discovered in the analyzed fields. Conclusions: Our positions present 1 arcsec precision with an accuracy of 0.2-0.4 arcsec, sufficient for achieving our immediate main goals. The observations and data reduction were conducted by our network members, which included some students and amateurs supervised by professional astronomers. In most cases, we increased the observational arcs decreasing the uncertainties in the orbits, while in some cases the new positions allowed us to recover some bodies endangered to be lost, defining their orbits. Based on observations acquired in Pic du Midi, Haute Provence, La Silla, Cerro Tololo, Las Campanas, Cerro Armazones, Bucharest Urseanu, and York University Observatories.Astrometric and photometric data are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/511/A40