In this paper we present the enhanced X-ray Timing and Polarimetry mission. eXTP is a space science mission designed to study fundamental physics under extreme conditions of density, gravity and magnetism. The mission aims at determining the equation of state of matter at supra-nuclear density, measuring effects of QED, and understanding the dynamics of matter in strong-field gravity. In addition to investigating fundamental physics, eXTP will be a very powerful observatory for astrophysics that will provide observations of unprecedented quality on a variety of galactic and extragalactic objects. In particular, its wide field monitoring capabilities will be highly instrumental to detect the electro-magnetic counterparts of gravitational wave sources. The paper provides a detailed description of: 1) The technological and technical aspects, and the expected performance of the instruments of the scientific payload; 2) The elements and functions of the mission, from the spacecraft to the ground segment.X-ray instrumentation, X-ray Polarimetry, X-ray Timing, Space mission: eXTP PACS number(s): 95.55. Ka, 95.85.Nv, 95.75.Hi, 97.60.Jd, 97.60.Lf