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The ALMA-ALPINE [CII] survey is aimed at characterizing the properties of a sample of normal star-forming galaxies (SFGs). The ALMA Large Program to INvestigate (ALPINE) features 118 galaxies observed in the [CII]-158 μ m line and far infrared (FIR) continuum emission during the period of rapid mass assembly, right after the end of the HI reionization, at redshifts of 4 <   z  <  6. We present the survey science goals, the observational strategy, and the sample selection of the 118 galaxies observed with ALMA, with an average beam minor axis of about 0.85″, or ∼5 kpc at the median redshift of the survey. The properties of the sample are described, including spectroscopic redshifts derived from the UV-rest frame, stellar masses, and star-formation rates obtained from a spectral energy distribution (SED) fitting. The observed properties derived from the ALMA data are presented and discussed in terms of the overall detection rate in [CII] and FIR continuum, with the observed signal-to-noise distribution. The sample is representative of the SFG population in the main sequence at these redshifts. The overall detection rate in [CII] is 64% for a signal-to-noise ratio (S/N) threshold larger than 3.5 corresponding to a 95% purity (40% detection rate for S / N  >  5). Based on a visual inspection of the [CII] data cubes together with the large wealth of ancillary data, we find a surprisingly wide range of galaxy types, including 40% that are mergers, 20% extended and dispersion-dominated, 13% compact, and 11% rotating discs, with the remaining 16% too faint to be classified. This diversity indicates that a wide array of physical processes must be at work at this epoch, first and foremost, those of galaxy mergers. This paper sets a reference sample for the gas distribution in normal SFGs at 4 <   z  <  6, a key epoch in galaxy assembly, which is ideally suited for studies with future facilities, such as the James Webb Space Telescope (JWST) and the Extremely Large Telescopes (ELTs).