Several tens of thousands of spectroscopic observations were obtained for both components of our neighbouring star system Centauri in order to measure precise radial velocities and ultimately to search for any potential Earth-mass planets in the system, including those which may lie within the respective habitable zone around either star. This large number of observations was necessary to achieve the precision required to be sensitive to the tiny signals caused by Earth-mass planets. Furthermore, a couple of thousand spectra were also recorded for the two single stars Pavonis and Ceti, as well as about 100 { 200 observations each for four double-lined spectroscopic binaries (HR 159, HR 913, HR 7578, HD 181958). In order to overcome the problem of spectral cross-contamination in the spectra of Centauri, a proven Doppler reduction code was expanded to include a variable amount of light coming from both stars in the modelling process for every observation. The amount of contamination was first estimated by means of a line index ratio for every observation before the observed spectrum was modelled using two stellar templates simultaneously with the amount of contamination as an additional fit parameter. The entire methodology of getting from the observed spectra to the final radial velocities is described in detail. The newly developed Doppler reduction code was then successfully applied to the observations of the four double-lined spectroscopic binaries and a detailed analysis of their radial velocities is presented. In two cases this method gave a RV precision sufficient to detect giant planets ( 20 􀀀 30ms􀀀1). While no planets were detected in orbit around these stars, the residuals of one system (HR 913) show systematic variations. A and B. The final radial velocity data sets of Cen A and B have an RMS slightly greater than 6ms􀀀1, but the short-term precision is about a factor of two better. Although several strong peaks were found in the periodograms, no planets could be unequivocally identified around either star. Upper mass limits for possible planets around Cen A and B were also computed. After a discussion of the radial velocity results, the reality of the putative discovery of an Earth-mass planet in a 3.24-d orbit around Cen B (Dumusque et al., 2012) was investigated, but the presence of this putative planet could not be confirmed. An analysis of the radial velocities of the two single stars Pav and Ceti, for which a long-term radial velocity precision of about 4:5ms􀀀1 was achieved, is also presented, providing some context for the analysis of the radial velocities of Cen