We present the first fully and uniformly sampled, spatially complete HI survey of the entire Magellanic System with high velocity resolution (Δv=1.0km/s), performed with the Parkes Telescope. Approximately 24 percent of the southern sky was covered by this survey on a ≃5' grid with an angular resolution of HPBW=14.1'. A fully automated data-reduction scheme was developed for this survey to handle the large number of HI spectra (1.5x10
6). The individual Hanning smoothed and polarization averaged spectra have an rms brightness temperature noise of σ=0.12K. The final data-cubes have an rms noise of σ
rms≃0.05K and an effective angular resolution of ≃16'. In this paper we describe the survey parameters, the data-reduction and the general distribution of the HI gas. The Large Magellanic Cloud (
LMC) and the Small Magellanic Cloud (
SMC) are associated with huge gaseous features - the
Magellanic Bridge, the
Interface Region, the
Magellanic Stream, and the
Leading Arm - with a total HI mass of M(HI)=4.87x10
8M
☉[d/55kpc]
2, if all HI gas is at the same distance of 55kpc. Approximately two thirds of this HI gas is located close to the Magellanic Clouds (
Magellanic Bridge and
Interface Region), and 25% of the HI gas is associated with the
Magellanic Stream. The
Leading Arm has a four times lower HI mass than the
Magellanic Stream, corresponding to 6% of the total HI mass of the gaseous features. We have analyzed the velocity field of the Magellanic Clouds and their neighborhood introducing a LMC-standard-of-rest frame. The HI in the
Magellanic Bridge shows low velocities relative to the Magellanic Clouds suggesting an almost parallel motion, while the gas in the
Interface Region has significantly higher relative velocities indicating that this gas is leaving the
Magellanic Bridge building up a new section of the
Magellanic Stream. The
Leading Arm is connected to the
Magellanic Bridge close to an extended arm of the
LMC. The clouds in the
Magellanic Stream and the
Leading Arm show significant differences, both in the column density distribution and in the shapes of the line profiles. The HI gas in the
Magellanic Stream is more smoothly distributed than the gas in the
Leading Arm. These morphological differences can be explained if the
Leading Arm is at considerably lower z-heights and embedded in a higher pressure ambient medium.