We present the ATLAS 3D Hi survey of a volume-limited, complete sample of 166 nearby early-type galaxies (ETGs) brighter than M K=-21.5. The survey is mostly based on data taken with the Westerbork Synthesis Radio Telescope, which enables us to detect Hi down to 5 × 10 6-5 × 10 7M ⊙ within the survey volume. We detect ~40per cent of all ETGs outside the Virgo galaxy cluster and ~10per cent of all ETGs inside it. This demonstrates that it is common for non-cluster ETGs to host Hi. The morphology of the detected gas varies in a continuous way from regular, settled Hi discs and rings to unsettled gas distributions (including tidal or accretion tails) and systems of clouds scattered around the galaxy. The majority of the detections consist of Hi discs or rings (1/4 of all ETGs outside Virgo) so that if Hi is detected in an ETG it is most likely distributed on a settled configuration. These systems come in two main types: small discs [M ⊙], which are confined within the stellar body and share the same kinematics of the stars; and large discs/rings [M(Hi) up to 5 × 10 9M ⊙], which extend to tens of kpc from the host galaxy and are in half of the cases kinematically decoupled from the stars. Neutral hydrogen seems to provide material for star formation in ETGs. Galaxies containing Hi within ~1R e exhibit signatures of on-going star formation in ~70per cent of the cases, approximately five times more frequently than galaxies without central Hi. The interstellar medium (ISM) in the centre of these galaxies is dominated by molecular gas, and in ETGs with a small gas disc the conversion of Hi into H 2 is as efficient as in spirals. The ETG Hi mass function is characterized by M*~ 2 × 10 9M ⊙ and by a slope α~-0.7. Compared to spirals, ETGs host much less Hi as a family. However, a significant fraction of all ETGs are as Hi-rich as spiral galaxies. The main difference between ETGs and spirals is that the former lack the high-column-density Hi typical of the bright stellar disc of the latter. The ETG Hi properties vary with environment density in a more continuous way than suggested by the known Virgo versus non-Virgo dichotomy. We find an envelope of decreasing M(Hi) and M(Hi)/L K with increasing environment density. The gas-richest galaxies live in the poorest environments (as found also with CO observations), where the detection rate of star formation signatures is higher. Galaxies in the centre of Virgo have the lowest Hi content, while galaxies at the outskirts of Virgo represent a transition region and can contain significant amounts of Hi, indicating that at least a fraction of them has joined the cluster only recently after pre-processing in groups. Finally, we find an Hi morphology-density relation such that at low environment density (measured on a local scale) the detected Hi is mostly distributed on large, regular discs and rings, while more disturbed Hi morphologies dominate environment densities typical of rich groups. This confirms the importance of processes occurring on a galaxy-group scale for the evolution of ETGs.