  We introduce VLT-MUSE observations of the central 2$'\times2'$ (30$\times$30
pc) of the Tarantula Nebula in the Large Magellanic Cloud. The observations
provide an unprecedented spectroscopic census of the massive stars and ionised
gas in the vicinity of R136, the young, dense star cluster located in NGC 2070,
at the heart of the richest star-forming region in the Local Group.
Spectrophotometry and radial-velocity estimates of the nebular gas
(superimposed on the stellar spectra) are provided for 2255 point sources
extracted from the MUSE datacubes, and we present estimates of stellar radial
velocities for 270 early-type stars (finding an average systemic velocity of
271$\pm$41 km/s). We present an extinction map constructed from the nebular
Balmer lines, with electron densities and temperatures estimated from intensity
ratios of the [SII], [NII], and [SIII] lines. The interstellar medium, as
traced by H$\alpha$ and [NII] $\lambda$6583, provides new insights in regions
where stars are probably forming. The gas kinematics are complex, but with a
clear bi-modal, blue- and red-shifted distribution compared to the systemic
velocity of the gas centred on R136. Interesting point-like sources are also
seen in the eastern cavity, western shell, and around R136; these might be
related to phenomena such as runaway stars, jets, formation of new stars, or
the interaction of the gas with the population of Wolf--Rayet stars. Closer
inspection of the core reveals red-shifted material surrounding the strongest
X-ray sources, although we are unable to investigate the kinematics in detail
as the stars are spatially unresolved in the MUSE data. Further papers in this
series will discuss the detailed stellar content of NGC 2070 and its integrated
stellar and nebular properties.
