We use high-resolution N-body/smoothed particle hydrodynamic simulations to study the hydrodynamical and gravitational interaction between the Large Magellanic Cloud (LMC) and the Milky Way Galaxy. We model the dark and hot extended halo components as well as the stellar/gaseous discs of the two galaxies. Both galaxies are embedded in extended cuspy ΛCDM dark matter haloes. We follow the previous 4 Gyr of the LMC's orbit such that it ends up with the correct location and orientation on the sky. Tidal forces elongate the LMC's disc, forcing a bar and creating a strong warp and diffuse stellar halo, although very few stars become unbound. The stellar halo may account for some of the microlensing events observed towards the LMC. Ram pressure from a low-density ionized halo is then sufficient to remove 1.4 × 108 M⊙ of gas from the LMC's disc, forming a great circle trailing stream around the Galaxy. The column density of stripped gas falls by two orders of magnitude 100 degrees from the LMC and the radial velocity along the trailing stream agrees well with the observations. The LMC does not induce any response in the Milky Way disc. On the contrary, the tides raised by the Milky Way determine the truncation of the satellite at about 11 kpc. After several gigayears of interaction, the gas disc of the LMC is smaller than the stellar disc due to ram pressure, and its size and morphology compare well with the observational data.