Context. We present radial-velocity measurements of eight stars observed with the HARPS Echelle spectrograph mounted on the 3.6-m telescope in La Silla (ESO, Chile). Data span more than ten years and highlight the long-term stability of the instrument. Aims: We search for potential planets orbiting HD 20003, HD 20781, HD 21693, HD 31527, HD 45184, HD 51608, HD 134060 and HD 136352 to increase the number of known planetary systems and thus better constrain exoplanet statistics. Methods: After a preliminary phase looking for signals using generalized Lomb-Scargle periodograms, we perform a careful analysis of all signals to separate bona-fide planets from signals induced by stellar activity and instrumental systematics. We finally secure the detection of all planets using the efficient MCMC available on the Data and Analysis Center for Exoplanets (DACE web-platform), using model comparison whenever necessary. Results: In total, we report the detection of twenty new super-Earth to Neptune-mass planets, with minimum masses ranging from 2 to 30 MEarth and periods ranging from 3 to 1300 days, in multiple systems with two to four planets. Adding CORALIE and HARPS measurements of HD20782 to the already published data, we also improve the characterization of the extremely eccentric Jupiter orbiting this visual companion of HD 20781. Based on observations made with the HARPS instrument on the ESO 3.6 m telescope at La Silla Observatory under the GTO program 072.C-0488 and Large program 193.C-0972/193.C-1005/.The analysis of the radial-velocity measurements were performed using the Data and Analysis Center for Exoplanets (DACE) developed in the frame of the Swiss NCCR PlanetS and available for the community at the following address: http://https://dace.unige.ch/.RV data are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/622/A37.