We know now from radial velocity surveys and transit space missions that planets only a few times more massive than our Earth are frequent around solar-type stars. Fundamental questions about their formation history, physical properties, internal structure, and atmosphere composition are, however, still to be solved. We present here the detection of a system of four low-mass planets around the bright (V = 5.5) and close-by (6.5 pc) star HD 219134. This is the first result of the Rocky Planet Search programme with HARPS-N on the Telescopio Nazionale Galileo in La Palma. The inner planet orbits the star in 3.0935 ± 0.0003 days, on a quasi-circular orbit with a semi-major axis of 0.0382 ± 0.0003 AU. Spitzer observations allowed us to detect the transit of the planet in front of the star making HD 219134 b the nearest known transiting planet to date. From the amplitude of the radial velocity variation (2.25 ± 0.22 ms^-1) and observed depth of the transit (359 ± 38 ppm), the planet mass and radius are estimated to be 4.36 ± 0.44 M_⊕ and 1.606 ± 0.086 R_⊕, leading to a mean density of 5.76 ± 1.09 g cm^-3, suggesting a rocky composition. One additional planet with minimum-mass of 2.78 ± 0.65 M_⊕ moves on a close-in, quasi-circular orbit with a period of 6.767 ± 0.004 days. The third planet in the system has a period of 46.66 ± 0.08 days and a minimum-mass of 8.94 ± 1.13 M_⊕, at 0.233 ± 0.002 AU from the star. Its eccentricity is 0.46 ± 0.11. The period of this planet is close to the rotational period of the star estimated from variations of activity indicators (42.3 ± 0.1 days). The planetary origin of the signal is, however, thepreferred solution as no indication of variation at the corresponding frequency is observed for activity-sensitive parameters. Finally, a fourth additional longer-period planet of mass of 71 M_⊕ orbits the star in 1842 days, on an eccentric orbit (e = 0.34 ± 0.17) at a distance of 2.56 AU.