Context. The Tarantula Nebula in the Large Magellanic Cloud is our closest view of a starburst region and is the ideal environment to investigate important questions regarding the formation, evolution and final fate of the most massive stars. Aims: We analyze the multiplicity properties of the massive O-type star population observed through multi-epoch spectroscopy in the framework of the VLT-FLAMES Tarantula Survey. With 360 O-type stars, this is the largest homogeneous sample of massive stars analyzed to date. Methods: We use multi-epoch spectroscopy and variability analysis to identify spectroscopic binaries. We also use a Monte-Carlo method to correct for observational biases. By modeling simultaneously the observed binary fraction, the distributions of the amplitudes of the radial velocity variations and the distribution of the time scales of these variations, we constrain the intrinsic current binary fraction and period and mass-ratio distributions. Results: We observe a spectroscopic binary fraction of 0.35 ± 0.03, which corresponds to the fraction of objects displaying statistically significant radial velocity variations with an amplitude of at least 20 km s-1. We compute the intrinsic binary fraction to be 0.51 ± 0.04. We adopt power-laws to describe the intrinsic period and mass-ratio distributions: f(log 10P/d) ~ (log 10P/d)π (with log 10P/d in the range 0.15-3.5) and f(q) ~ qκ with 0.1 ≤ q = M2/M1 ≤ 1.0. The power-law indexes that best reproduce the observed quantities are π = -0.45 ± 0.30 and κ = -1.0 ± 0.4. The period distribution that we obtain thus favours shorter period systems compared to an Öpik law (π = 0). The mass ratio distribution is slightly skewed towards low mass ratio systems but remains incompatible with a random sampling of a classical mass function (κ = -2.35). The binary fraction seems mostly uniform across the field of view and independent of the spectral types and luminosity classes. The binary fraction in the outer region of the field of view (r > 7.8', i.e. ≈117 pc) and among the O9.7 I/II objects are however significantly lower than expected from statistical fluctuations. The observed and intrinsic binary fractions are also lower for the faintest objects in our sample (Ks > 15.5 mag), which results from observational effects and the fact that our O star sample is not magnitude-limited but is defined by a spectral-type cutoff. We also conclude that magnitude-limited investigations are biased towards larger binary fractions. Conclusions: Using the multiplicity properties of the O stars in the Tarantula region and simple evolutionary considerations, we estimate that over 50% of the current O star population will exchange mass with its companion within a binary system. This shows that binary interaction is greatly affecting the evolution and fate of massive stars, and must be taken into account to correctly interpret unresolved populations of massive stars. Based on observations collected at the European Southern Observatory under program ID 182.D-0222.Full Tables 1-3 are only available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr(130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/550/A107Appendices are available in electronic form at http://www.aanda.org