Massive binaries (MBs) play a crucial role in the Universe. Knowing the distributions of their orbital parameters (OPs) is important for a wide range of topics, from stellar feedback to binary evolution channels, from the distribution of supernova types to gravitational wave progenitors, yet, no direct measurements exist outside the Milky Way. The Tarantula Massive Binary Monitoring was designed to help fill this gap by obtaining multi-epoch radial velocity monitoring of 102 MBs in the 30 Dor. In this paper, we analyse 32 VLT/FLAMES observations of 93 O- and 7 B-type binaries. We performed a Fourier analysis and obtained orbital solutions for 82 systems: 51 single- and 31 double-lined spectroscopic binaries. Overall, the OPs and binary fraction are remarkably similar across the 30 Dor region and compared to existing Galactic samples (GSs). This indicates that within these domains environmental effects are of second order in shaping the properties of MBs. A small difference is found in the distribution of orbital periods (OrbPs), which is slightly flatter (in log space) in 30 Dor than in the Galaxy, although this may be compatible within error estimates and differences in the fitting methodology. Also, OrbPs in 30 Dor can be as short as 1.1 d; somewhat shorter than seen in GSs. Equal mass binaries q>0.95 in 30 Dor are all found outside NGC 2070 the very young and massive cluster at 30 Dors core. One outstanding exception however is the fact that earliest spectral types (O2-O7) tend to have shorter OrbPs than latter (O9.2-O9.7). Our results point to a relative universality of the incidence rate of MBs and their OPs in the metallicity range from solar ($Z_{odot}$) to about $0.5Z_{odot}$. This provides the first direct constraints on MB properties in massive star-forming galaxies at the Universes peak of star formation at redshifts z~1 to 2, which are estimated to have $Z~0.5Z_{odot}$.
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