The ALMA-ALPINE [CII] survey (A2C2S) aims at characterizing the properties of normal star-forming galaxies (SFGs) observed in the [CII]-158μm line in the period of rapid mass assembly at redshifts 4 < z < 6. Here we present the survey and the selection of 118 galaxies observed with ALMA, selected from large samples of galaxies with spectroscopic redshifts derived from UV-rest frame. The observed properties derived from the ALMA data are presented and discussed in terms of the overall detection rate in [CII] and far-IR continuum. The sample is representative of the SFG population at these redshifts. The overall detection rate is 61% down to a flux limit of 0.07 mJy. From 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 32.4% mergers, 25.7% extended and dispersion dominated, 13.5% rotating discs, and 16.2% compact, the remaining being too faint to be classified. ALPINE sets a reference sample for the gas distribution in normal star-forming galaxies at a key epoch in galaxy assembly, ideally suited for studies with future facilities like JWST and ELTs.